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BIRATIONAL GEOMETRY OF SINGULAR MODULI SPACES OF

O’GRADY TYPE

CIARAN MEACHAN AND ZIYU ZHANG

Abstract. Following Bayer and Macrı, we study the birational geometry of singular modulispaces M of sheaves on a K3 surface X which admit symplectic resolutions. More precisely,we use the Bayer-Macrı map from the space of Bridgeland stability conditions Stab(X) to thecone of movable divisors on M to relate wall-crossing in Stab(X) to birational transformationsof M . We give a complete classification of walls in Stab(X) and show that every minimalbirational model of M in the sense of the log minimal model program appears as a modulispace of Bridgeland semistable objects on X. An essential ingredient of our proof is an isometrybetween the orthogonal complement of a Mukai vector inside the algebraic Mukai lattice ofX and the Neron-Severi lattice of M which generalises results of Yoshioka, as well as Peregoand Rapagnetta. Moreover, this allows us to conclude that the symplectic resolution of Mis deformation equivalent to the 10-dimensional irreducible holomorphic symplectic manifoldfound by O’Grady.

1. Introduction

1.1. Background. Let X be a complex projective K3 surface, v ∈ H∗alg(X,Z) a Mukai vector

and H an ample line bundle which is generic with respect to v (in the sense of [HL10, Section4.C] and [Yos01, Section 1.4]). We can always write the Mukai vector as a multiple of a primitiveclass, say v = mvp. If we further assume that v2 > 0 with respect to the Mukai pairing, thenthere is a precise classification of moduli spaces MH(v) of Gieseker H-semistable sheaves on Xwith Mukai vector v which goes as follows:

• If m = 1, then v is a primitive Mukai vector and MH(v) is smooth. Moreover, inhis seminal paper [Muk84], Mukai showed that MH(v) is an irreducible holomorphicsymplectic manifold, parametrising H-stable sheaves.

• If m > 2, then MH(v) has symplectic singularities (in the sense of [Bea00]). Its smoothlocus parametrises H-stable sheaves, while its singular locus parametrises S-equivalenceclasses of strictly semistable sheaves. This case splits into two radically different situa-tions:

– If m = 2 and v2p = 2, then MH(v) has a symplectic resolution; see [O’G99, LS06].

– If m > 2 or v2p > 2, then MH(v) does not admit a symplectic resolution; see

[KLS06].

The geometry of smooth moduli spaces MH(v), or irreducible holomorphic symplectic manifoldsin general, has been the subject of intensive study for many years. In particular, there are manyresults about their birational geometry in the literature; see [Huy03, HT09] for instance. Onthe other hand, Bridgeland [Bri08, Section 14] showed that these Gieseker moduli spaces can

2010 Mathematics Subject Classification. 14D20 (Primary); 14F05, 14J28, 18E30 (Secondary).Key words and phrases. Bridgeland stability conditions, derived categories, moduli spaces of sheaves and

complexes, wall crossing, symplectic resolutions.

1

2 CIARAN MEACHAN AND ZIYU ZHANG

be realised as moduli spaces of σ-semistable objects in the ‘large volume limit’ of (a certainconnected component Stab†(X) of) his stability manifold Stab(X). More precisely, he provedthat the stability manifold comes with a wall and chamber decomposition in the sense that theset of σ-semistable objects with some fixed numerical invariants is constant in each chamberand an object of the bounded derived category D(X) of coherent sheaves on X can only becomestable or unstable by crossing a wall, that is, a real codimension one submanifold of Stab†(X).Furthermore, Bridgeland conjectured that crossing a wall should induce a birational transfor-mation between the corresponding moduli spaces and this vision was recently crystallised in arevolutionary paper [BM14a] by Bayer and Macrı.

Before going any further, we should say that these ideas concerning wall-crossing have beensuccessfully applied to the study of the birational geometry of moduli spaces of sheaves onabelian surfaces; see [Yos09, MYY11, Mea12, Mac12, Yos12, AB13, MM13, MYY13, YY14].However, the focus of this paper will be on moduli spaces of sheaves on K3 surfaces as above.

For any Mukai vector v with v2 > 0, on each moduli space of Bridgeland semistable objectsMσ(v), Bayer and Macrı used the classical technique of determinant line bundles to constructan ample line bundle `σ on Mσ(v). In particular, this gives rise to a ‘linearisation map’ fromany chamber C in Stab†(X) to the nef cone of the moduli space Nef(MC(v)) corresponding tothat chamber. This map is an essential part of this paper; we call it the Bayer-Macrı map.

Moreover, when the Mukai vector v is primitive, they obtained a complete picture relating wallcrossing on Stab†(X) to the birational geometry of the corresponding moduli spaces. Theystudied how the moduli space changes when crossing a wall and classified all the walls in termsof the Mukai lattice. In particular, they verified the conjecture of Bridgeland, by showing thatmoduli spaces corresponding to two neighbouring chambers are indeed birational. For each typeof wall, they could produce an explicit birational map, which identifies the two moduli spacesaway from loci of codimension at least two and hence identifies the Neron-Severi groups of them.Using these birational maps, they could glue the linearisation maps defined on each chambertogether to get a global continuous Bayer-Macrı map from Stab†(X) to the Neron-Severi group ofany generic moduli space Mσ(v) and prove that, via wall crossings on Stab†(X), every birationalminimal model of Mσ(v) appears as a Bridgeland moduli space.

1.2. Summary of main results. This paper grew out of an attempt to first understand[BM14b, BM14a] and then generalise their techniques to the simplest singular case in the hopeof obtaining similar results. In particular, we are interested in the case when v = 2vp and vp isa primitive Mukai vector with square equal to 2 with respect to the Mukai pairing. We say thatsuch a Mukai vector v is of O’Grady type. The benefit of considering this type of Mukai vectorv is that we will be able to show the existence of a symplectic resolution of the moduli spaceMσ(v) under any generic stability condition σ, which will allow us to reuse many arguments in[BM14a]. Our first main result is the following

Theorem (5.4). Let X be a projective K3 surface and v be a Mukai vector of O’Grady type.For any two generic stability conditions σ, τ ∈ Stab†(X) with respect to v, there is a birationalmap Φ∗ : Mσ(v) 99K Mτ (v) induced by a derived (anti-)autoequivalence1 on D(X), which is anisomorphism in codimension one.

1An anti-autoequivalence is an equivalence of categories from D(X)op to D(X), which takes every exact trianglein D(X) to an exact triangle in D(X), but with all the arrows reversed. All examples of this notion encountered inthis paper are compositions of autoequivalences with the derived dual functor RHom(−,OX); see the discussionafter [BM14a, Theorem 1.1].

BIRATIONAL GEOMETRY OF SINGULAR MODULI SPACES OF O’GRADY TYPE 3

This result is a generalisation of [BM14a, Theorem 1.1]. Although the statements are quitesimilar, the singular version is more involved. For instance, in the primitive case, every modulispace under a generic stability condition is smooth and has trivial canonical class. Therefore,any birational map between two such moduli spaces naturally extends to an isomorphism incodimension one [GHJ03, Proposition 21.6]. However, this does not hold in the singular case.

To fix this issue, we introduce the notation of a stratum preserving birational map between twosingular moduli spaces of O’Grady type and show that the behavior of such birational maps incodimension one is as nice as it is in the smooth world; see Proposition 2.6.

A refinement of Theorem 5.4, which generalises [BM14a, Theorem 1.2], is the following

Theorem (7.6). Let v be a Mukai vector of O’Grady type and σ ∈ Stab†(X) be any genericstability condition with respect to v. Then

(1) We have a globally defined continuous Bayer-Macrı map ` : Stab†(X) → NS(Mσ(v)),which is independent of the choice of σ. Moreover, for any generic stability conditionτ ∈ Stab†(X), the moduli space Mτ (v) is the birational model corresponding to `τ .

(2) If C ⊂ Stab†(X) is the open chamber containing σ, then `(C) = Amp(Mσ(v)).

(3) The image of ` is equal to Big(Mσ(v)) ∩ Mov(Mσ(v)). In particular, every K-trivialQ-factorial birational model of Mσ(v) which is isomorphic to Mσ(v) in codimension 1appears as a moduli space Mτ (v) for some generic stability condition τ ∈ Stab†(X).

In our situation, Mσ(v) is a K-trivial Q-factorial variety with canonical (hence log-terminal)singularities. Therefore Mσ(v) (together with an empty divisor) is a log minimal model. For suchsymplectic varieties which admit symplectic resolutions, the existence and termination of log-flips have been established in [BCHM10, Corollary 1.4.1] and [LP14, Theorem 4.1] respectively.Therefore the log minimal model program works in this case. In particular, every K-trivial Q-factorial birational model of Mσ(v) which is isomorphic to it in codimension 1 (in other words,every log minimal model which is log-MMP related to Mσ(v)) can be obtained through a finitesequence of log-flops. Theorem 7.6 shows that every such log minimal model has an interpretionas a Bridgeland moduli space Mτ (v) for some generic stability condition τ . This picture isparallel to [BM14a, Theorem 1.2] which deals with the case of primitive Mukai vectors.

As an application of Theorem 7.6, we can also formulate a Torelli-type theorem for singularmoduli spaces of O’Grady type, which is parallel to [BM14a, Corollary 1.3]; see Corollary 7.8for more details.

The proof of Theorem 5.4 and Theorem 7.6 relies on a complete classification of walls, asstated in Theorem 5.1 and Theorem 5.3, which generalises [BM14a, Theorem 5.7]. Althoughour proofs follow their approach very closely, the technical details are much more involved.The main difficulty is that the proof of [BM14a, Theorem 5.7] uses many results on irreducibleholomorphic symplectic manifolds which are not available to us in the singular world. Eachtime they use an argument which relies on smoothness in an implicit way, we have to either finda way around it or prove a new version that works in the singular case. One instance of this,which leads to an interesting by-product of the project, goes as follows.

The main ingredient for the Bayer-Macrı map is the classical construction of determinant linebundles on MH(v); see [HL10, Section 8.1]. Its algebraic version, which is often referred toas the Mukai morphism, is a map of lattices θσ : v⊥ → NS(Mσ(v)) where the orthogonalcomplement is taken in the algebraic cohomology H∗alg(X,Z). When v is primitive, Yoshioka


proved in [Yos01] that θσ is an isometry with respect to the Mukai pairing on H∗alg(X,Z) and

the Beauville-Bogomolov pairing on NS(Mσ(v)).

However, when v is a Mukai vector of O’Grady type, such a theorem for Bridgeland modulispaces does not seem to exist in literature. In fact, to make sense of it, we have to find awell-defined pairing on the Neron-Severi lattice of the moduli space in the first place. Luckily, aspecial case of it, on the level of cohomology, concerning only Gieseker moduli spaces, was provedin [PR13]. Using their approach, we can similarly define the bilinear pairing on NS(Mσ(v)), ormore generally on H2(Mσ(v)). Although their proof of the isometry does not adapt immediatelyto the case of Bridgeland moduli spaces, all the key ideas are there and we are able to deducethe desired result using Fourier-Mukai transforms and deformation theory.

A necessary and crucial step in this project, as well as an interesting result in its own right, is theanalogous result concerning the cohomological version of the Mukai morphism in the singularsetting. In particular, we have

Theorem (2.7). Let v be a Mukai vector of O’Grady type and σ ∈ Stab†(X) be a generic stability

condition with respect to v. Denote Mσ(v) by M and its symplectic resolution by π : M → M .Then

(1) The pullback map π∗ : H2(M,Z) → H2(M,Z) is injective and compatible with the(mixed) Hodge structures. In particular, H2(M,Z) carries a pure Hodge structure ofweight two, and the restriction of the Beauville-Bogomolov quadratic form q(−,−) on

H2(M,Z) defines a quadratic form q(−,−) on H2(M,Z).

(2) There exists a well-defined Mukai morphism θtrσ : v⊥,tr → H2(M,Z) induced by the

(quasi-)universal family over M stσ (v), which is a Hodge isometry.

We point out that v⊥,tr in the above theorem denotes the orthogonal complement in the totalcohomology H∗(X,Z).

If σ lies in the Gieseker chamber then Theorem 2.7 is precisely [PR13, Theorem 1.7]. Themore general statement follows by combining the ideas of Perego and Rapagnetta with somenew ingredients inspired by [BM14a]. More precisely, for an arbitrary σ ∈ Stab†(X), we use aFourier-Mukai transform to identify Mσ(v) with a moduli space of twisted sheaves on anotherK3 surface. Then we deform the underlying twisted K3 surface to an untwisted K3 surfacewhere the isometry has been proved in [PR13, Theorem 1.7]. The observation in [PR13] thatthis isometry is preserved under these two operations finishes the proof. However, the presenceof a non-trivial Brauer class makes the deformation argument more complicated. For instance,the issue of ampleness caused by (−2)-classes and genericness of the polarisation, turn out tobe far from straightforward.

As an immediate consequence, we obtain the algebraic version of the Theorem 2.7 as follows.

Corollary (2.8). Under the assumptions of Theorem 2.7, we have

(1) Lefschetz (1, 1) theorem holds for M . That is, NS(M) = H1,1(M,Z);

(2) The restriction of the pullback map π on NS(X) is an injective map π∗ : NS(M) →NS(M), which is compatible with the Beauville-Bogomolov pairings q(−,−) and q(−,−);

(3) The restriction of the Mukai morphism θtrσ on the algebraic Mukai lattice is an isometry

θσ : v⊥ → NS(M). In particular, q(−,−) is a non-degenerate pairing on NS(M) withsignature (1, ρ(X)).


We remark that it is only the third statement in the above corollary which is needed in ourstudy of the Bayer-Macrı map. However, we do not know a direct proof of this statement whichdoes not utilise the cohomological Mukai morphism; the reason being that our proof involvesa deformation argument. Unlike cohomology groups which stay constant in a family as a localsystem, the Neron-Severi groups do not behave well under deformations.

As a consequence of Theorem 2.7, we obtain a generalisation of [PR13, Theorem 1.6] fromGieseker moduli spaces to Bridgeland moduli spaces. The proof combines the arguments in[PR13] and the deformation techniques developed in Section 3 below.

Corollary (3.16). Let v be a Mukai vector of O’Grady type and σ ∈ Stab†(X) be any genericstability condition with respect to v. Then the symplectic resolution of Mσ(v) is deformationequivalent to the irreducible holomorphic symplectic manifold constructed by O’Grady in [O’G99].

This result in particular implies that, by resolving singular moduli spaces of Bridgeland semistableobjects on K3 surfaces, we cannot get any new deformation types of irreducible holomorphicsymplectic manifolds other than the one discovered by O’Grady in [O’G99]. It is somewhatdisappointing, but supports the long-standing belief that deformation types of irreducible holo-morphic symplectic manifolds are rare.

1.3. Outline of the paper. In Section 2, we collect together the necessary properties of modulispaces of O’Grady type that we will need. After briefly mentioning their stratifications andresolutions, we study birational maps between them and state Theorem 2.7. We also give a briefaccount of various cones of divisors on these moduli spaces.

Section 3 is devoted to the proof of Theorem 2.7. We provide some lemmas on the deformationsof twisted polarised K3 surfaces, as well as on the existence of local relative moduli spaces. Themain proof comes after all these lemmas.

In Section 4, we briefly review the Bayer-Macrı map constructed in [BM14b], but only from theaspect that will become important in our later discussion. We also use the Bayer-Macrı mapto generalise the ampleness results proved in [BM14b]. We refer interested readers to [BM14b,Section 3] for the original construction, which provides a very conceptual way to understand thepositivity lemma there.

We state our classification theorems of potential walls in Section 5; see Theorems 5.1 and 5.3.We also establish the birational maps relating moduli spaces for two chambers separated by awall; see Theorem 5.4. In this section we only state the results, while leaving all proofs for nextsection. It is worth pointing out that our first classification theorem is true for arbitrary Mukaivectors, while the second classification theorem only works for Mukai vectors of O’Grady type.

Section 6 is devoted to the proof of all results in the section 5. We try to make our proofs shortand avoid repeating any existing arguments by making many references to results in [BM14a,Section 6-9]. Readers who are not interested in proofs can safely skip this section withoutaffecting the coherence of logic.

In Section 7, we describe and prove our main result. In particular, Theorem 7.6 provides a preciserelationship between wall crossings on the stability manifold and the birational geometry of thecorresponding moduli spaces.

Our main references of the paper are [BM14b, BM14a]. Since our presentation closely followstheirs, all background knowledge required for this paper are already included in the first fewsections of those. Nevertheless, we recommend readers the following references for generalknowledge of some relevant topics: [HL10] for moduli spaces of sheaves, [GHJ03, Part 3] for


geometry of irreducible holomorphic symplectic manifolds, [Bri08] for stability conditions on K3surfaces, and [Cal00] for twisted sheaves.

Acknowledgements: We are most grateful to Arend Bayer and Alastair Craw for all their help,support and encouragement throughout this project. Special thanks to Daniel Huybrechts forhis invaluable advice and guidance at various stages of this work. We also thank Christian Lehn,Sonke Rollenske and Kota Yoshioka for kindly answering our questions, and the referees for theirvery helpful comments. Z. Z. would also like to thank Jun Li for his initial suggestion of lookinginto this topic. C. M. is supported by an EPSRC Doctoral Prize Research Fellowship GrantEP/K503034/1 and Z. Z. is supported by an EPSRC Standard Research Grant EP/J019410/1.We also appreciate the support from the University of Bonn, the Max Planck Institute forMathematics, and the SFB/TR-45 during the initial stage of this project as well as the HausdorffResearch Institute for Mathematics for its conclusion.

2. Singular Moduli Spaces of O’Grady Type

We start our discussion by collecting some useful properties of the singular moduli spaces ofO’Grady type, which will be the central geometric objects that are studied in the whole paper.We will see that despite of the singularities, these moduli spaces behave very similar to smoothmoduli spaces, in the sense that many nice properties of smooth moduli spaces can be generalisedto these with some extra care of the singular loci. After introducing necessary notations andbackground materials, we will mainly focus on two aspects of these moduli spaces: birationalmaps between them and Mukai morphisms on their cohomology. After that, we will brieflymention various cones of divisors on singular moduli spaces of O’Grady type.

2.1. Moduli spaces and symplectic resolutions. We start by recalling the basic notion ofa Bridgeland moduli space. Let X be a projective K3 surface and v ∈ H∗alg(X,Z) be a Mukai

vector. Throughout this paper we will always assume v2 > 0. Moreover, there is a unique wayto write v = mvp for some positive integer m and primitive class vp ∈ H∗alg(X,Z). We say m

is the divisibility and vp is the primitive part of v. Let σ ∈ Stab†(X) be a Bridgeland stabilitycondition in the distinguished component of the stability manifold. The stability manifoldStab†(X) comes with a wall and chamber structure with respect to v as described above (seealso [Bri08, Section 9] and [BM14b, Proposition 2.3]), and we say σ ∈ Stab†(X) is generic if itdoes not lie on any wall.

It was proven in [BM14b, Theorem 1.3] (which generalises [MYY13, Theorem 0.0.2]) that, for astability condition σ ∈ Stab†(X) which is generic with respect to v, there exists a coarse modulispace MX,σ(v), which parametrises the S-equivalence classes of σ-semistable objects of class von X. Furthermore, it is a normal projective irreducible variety with Q-factorial singularities.By [BM14a, Theorem 2.15] (or originally [Yos01, Tod08]), MX,σ(v) is non-empty, and a genericpoint of it represents a σ-stable object.

A classical theorem, originally proved by Mukai in [Muk84] (see also [BM14b, Theorem 6.10] and[BM14a, Theorem 3.6]), says that when σ is generic and v is primitive, the moduli space MX,σ(v)is an irreducible holomorphic symplectic manifold, which parametrises σ-stable objects of classv only. However, when v is non-primitive, it is proved in [KLS06, Theorem 6.2] for Giesekermoduli spaces and [BM14a, Theorem 3.10] for Bridgeland moduli spaces, that MX,σ(v) hassymplectic singularities.

For simplicity, we often drop the K3 surface X from the notation when it is clear from thecontext. In fact, the moduli space only depends on the choice of the chamber C containing σ.


Therefore we sometimes also denote the moduli space for any stability condition contained in achamber C by MX,C(v), or simply MC(v). For the convenience of later discussion, we also makethe following definition.

Definition 2.1. We say a Mukai vector v ∈ H∗alg(X,Z) is of O’Grady type if it can be written

as v = mvp, where m = 2 and v2p = 2. We say a moduli space MX,σ(v) is of O’Grady type if σ

is a generic stability condition, and v is a Mukai vector of O’Grady type.

The importance of this particular type of moduli spaces lies in the study of symplectic resolutionsof MX,σ(v). In [O’G99], O’Grady constructed a symplectic resolution of a Gieseker modulispace with Mukai vector v = (2, 0,−2) and showed that it was not deformation equivalent toany existing example of homomorphic symplectic manifolds at the time. In [LS06], Lehn andSorger generalised the result to arbitrary Gieseker moduli spaces of O’Grady type, and gavea slightly different description of their symplectic resolutions. It was proved in [PR13] thatall these symplectic resolutions are in fact deformation equivalent to the one constructed byO’Grady. In [KLS06], it was proved that for a generic Gieseker moduli space with any othernon-primitive Mukai vector, a symplectic resolution does not exist.

By using the techniques developed in [BM14b, Section 7], we can easily generalise the existenceof symplectic resolutions to Bridgeland moduli spaces with the following proposition. Later inCorollary 3.16, we will show that all these symplectic resolutions are still deformation equivalentto the one constructed by O’Grady, hence do not provide new deformation type of irreducibleholomorphic symplectic manifolds.

Proposition 2.2. Let X be a projective K3 surface and v = mvp ∈ H∗alg(X,Z) be a Mukai

vector with m > 2, vp primitive and v2p > 0. Let σ ∈ Stab†(X) be generic with respect to v.

Then

• If m = 2 and v2p = 2, then Mσ(v) admits a symplectic resolution;

• If m > 2 or v2p > 2, then Mσ(v) does not admit a symplectic resolution.

Proof. By [BM14b, Lemma 7.3] and the discussion after that, there exists a twisted K3 surface(Y, α) where α ∈ Br(Y ) and a derived equivalence Φ : D(X) → D(Y, α) in the form of aFourier-Mukai transform, such that Φ induces an isomorphism MX,σ(v) ∼= MY,α,H(−Φ(v)),where MY,α,H(−Φ(v)) is the moduli space of α-twisted Gieseker H-semistable locally free sheaveson Y . Hence it suffices to prove the claims for twisted Gieseker moduli spaces. Therefore withoutloss of generality, we replace the Bridgeland moduli space in question by a twisted Giesekermoduli space MX,α,H(v).

By [Lie07, Proposition 2.3.3.6], there is a GIT construction for the twisted Gieseker modulispaces, which is precisely the same as in the case of untwisted Gieseker moduli spaces. Andby [Lie07, Proposition 2.2.4.9], the local deformation theory of twisted sheaves is also the sameas that of untwisted sheaves. Therefore the argument in [LS06] shows that MX,α,H(v) admitsa symplectic resolution as the blowup of its singular locus when m = 2 and v2

p = 2. Andthe argument in [KLS06] shows that MX,α,H(v) has no symplectic resolution when m > 2 orv2p > 2.

The existence of symplectic resolutions is critical for most results in the present paper. Thefollowing proposition is such an example. One property of irreducible holomorphic symplecticmanifolds used in [BM14a] is that, for a divisorial contraction on an irreducible holomorphicsymplectic manifold, the image of the contracted divisor has codimension exactly two, which is a


special case of [Nam01, Proposition 1.4], [Wie03, Theorem 1.2] or [Kal06, Lemma 2.11]. For ourpurpose, we need a version of this result in singular case. Thanks to the existence of symplecticresolutions, the same result can be easily proved for moduli spaces of O’Grady type. But wenevertheless state it under a more general setup as follows.

Proposition 2.3. Let M be a variety with symplectic singularities of dimension 2n admitting asymplectic resolution, and let N be a normal projective variety. Let ϕ : M → N be a birationalprojective morphism. We denote by Si the set of points p ∈ N such that dimϕ−1(p) = i. ThendimSi 6 2n−2i. In particular, if ϕ contracts a divisor D ⊂M , then we have dimϕ(D) = 2n−2.

Proof. The statement in question is equivalent to the following statement: if V ⊂ M is anyclosed subvariety of dimension 2n − i, then dimϕ(V ) > 2n − 2i; see for instance the proof of[Nam01, Proposition 1.4]. Without loss of generality, it suffices to prove this statement with theextra assumption that V is irreducible with the generic point ξV .

Let the symplectic resolution of M be π : M →M . We denote the closure of π−1(ξV ) in M by V .

Then we have π(V ) = V , hence dim V > dimV = 2n− i. Then we apply [Nam01, Proposition

1.4], [Wie03, Theorem 1.2] or [Kal06, Lemma 2.11] on the composition ϕ π : M → N and

conclude dimϕ(V ) = dim(ϕ π)(V ) > 2n− 2i.

However, we remind the readers that although many of our results are proved in the context ofdefinition 2.1, some of our results do work in more general situations. We will state very clearlywhich assumptions are made in every result.

2.2. Stratum preserving birational maps. Whenever σ is generic, there is a stratificationof the moduli space MX,σ(v) given by locally closed strata. The stable locus M st

X,σ(v), whichagrees with the smooth locus, is the unique open stratum. All the other lower dimensional strataare formed by lower dimensional moduli spaces. We refer the readers to the proof of [BM14a,Theorem 2.15] for the general case. Here we only describe the stratification for moduli spacesof O’Grady type.

For a moduli space of O’Grady type MX,σ(v), there is a chain of closed subschemes as follows:

MX,σ(vp) ⊂ Sym2MX,σ(vp) ⊂MX,σ(v), (2.1)

where the first inclusion is given by the diagonal morphism, and the second inclusion givesprecisely the strictly semistable locus of MX,σ(v), which agrees with the singular locus. Thischain of inclusions decomposes MX,σ(v) into the disjoint union of three locally closed strata.

More precisely, M stX,σ(v) = MX,σ(v)\ Sym2MX,σ(vp) parametrises all σ-stable objects of class v.

Every point in Sym2MX,σ(vp) represents the S-equivalent class containing a polystable objectE1 ⊕ E2, where E1, E2 ∈ MX,σ(vp) are both σ-stable objects of class vp. Such a point lies inthe diagonal MX,σ(vp) if and only if E1 and E2 are isomorphic.

With this stratification at hand, we are now ready to discuss birational maps between modulispaces of O’Grady type, which are compatible with the above stratifications.

When talking about birational maps between singular moduli spaces of O’Grady type, we empha-sise a special class of them, which preserve the natural stratifications described above. Almostall birational maps between these moduli spaces which occur in this paper belong to this class.Although the definition could be made for arbitrary moduli spaces under generic stability con-ditions, for the purpose of this paper, we restrict ourselves to moduli spaces of O’Grady type asfollows.


Definition 2.4. Let f : MX1,σ1(v1) 99K MX2,σ2(v2) be a birational map between two moduli

spaces of O’Grady type. Let MXi,σi(vi,p) ⊂ Sym2MXi,σi(vi,p) ⊂ MXi,σi(vi) be the standardstratification (2.1) for i = 1, 2, where vi,p denotes the primitive part of vi. If f is defined on thegeneric point of each stratum in MX1,σ1(v1), and takes each such generic point to the genericpoint of the corresponding stratum in MX2,σ2(v2), then we say that f is a stratum preservingbirational map.

Derived (anti-)equivalences are a very natural and particularly rich resource of stratum preserv-ing birational maps. The following lemma gives a criterion for such a derived (anti-)equivalenceto induce a stratum preserving birational map.

Lemma 2.5. Let Φ : D(X1) → D(X2) be a derived (anti-)equivalence and let MX1,σ1(v1) andMX2,σ2(v2) be two moduli spaces of O’Grady type. Assume that Φ induces a birational mapΦ∗ : MX1,σ1(v1) 99K MX2,σ2(v2). Then Φ∗ is stratum preserving if and only if the followingcondition holds: there exist a σ1-stable object E of class v1 and a σ1-stable object Ep of classv1,p, such that Φ(E) and Φ(Ep) are σ2-stable objects of classes v2 and v2,p respectively.

Proof. The necessity is part of the definition of f being stratum preserving, so we only discusssufficiency. By the openness of stability in [BM14a, Theorem 4.2] (which was originally provedin [Tod08]), the assumptions imply that the induced birational map Φ∗ takes the generic pointsof the open stratum M st

X1,σ1(v1) and the closed stratum MX1,σ1(v1,p) to the generic points of

corresponding strata in MX2,σ2(v2). It remains to show that Φ∗ takes the generic point of the

singular locus Sym2MX1,σ1(v1,p) to the generic point of Sym2MX2,σ2(v2,p). In fact, a generic

point Es in Sym2MX1,σ1(v1,p) can be represented by any extension of two generic stable objectsof class v1,p. Since Φ is a derived (anti-)equivalence, it preserves extensions (or switches thedirection). Hence Φ(Es) is again the extension of two generic stable objects of class v2,p, whichrepresents a generic point in Sym2MX2,σ2(v2,p), as desired.

A big advantage of stratum preserving birational maps is that they behave very much likebirational maps between smooth symplectic varieties. For example, the following propositiongeneralises a classical result about a birational map between two K-trivial smooth varieties, forinstance, in [GHJ03, Proposition 21.6].

Proposition 2.6. Let f : MX1,σ1(v1) 99K MX2,σ2(v2) be a stratum preserving birational mapbetween two moduli spaces of O’Grady type which is induced by a derived (anti-)equivalenceΦ : D(X1)→ D(X2). Furthermore, assume that there exists an open subset U ⊂M st

X1,σ1(v1) with

complement of codimension at least two, such that the restriction f |U is an injective morphismf |U : U →M st

X2,σ2(v2). Then f(U) has complement of codimension at least two in M st

X2,σ2(v2).

Proof. Since f is a stratum preserving birational map, there exists an open subset Us ⊂Sym2MX1,σ1(v1,p), such that f |Us : Us → Sym2MX2,σ2(v2,p) is an injective morphism. Now

we take Z1 to be the closure of M stX1,σ1

(v1)\U in MX1,σ1(v1), Z2 to be Sym2MX1,σ1(v1,p)\Us,and Z3 to be the closed stratum MX1,σ1(v1,p). We consider V = MX1,σ1(v1)\(Z1 ∪ Z2 ∪ Z3).Then it is easy to see that V is an open subset of MX1,σ1(v1). Moreover, V is the union ofU and an open subset of Us, hence has a complement of codimension at least two, and therestriction f |V is an injective morphism f |V : V →MX2,σ2(v2). Since f is induced by a derived(anti-)equivalence Φ, we see that Φ−1 defines an inverse of f on f(V ), and therefore f is anisomorphism from V to its image f(V ). Note that since V has no intersection with the closedstratum MX1,σ1(v1,p), f(V ) also has no intersection with the closed stratum MX2,σ2(v2,p).


For i = 1, 2, we write πi : Mi →MXi,σi(vi) for the symplectic resolution constructed in [O’G99,

LS06]. The construction there implies that both π1 : π−11 (V )→ V and π2 : π−1

2 (f(V ))→ f(V )are exactly the blowups of the singular loci. Therefore, the isomorphism f : V → f(V ) induces

another isomorphism f : π−11 (V )→ π−1

2 (f(V )). In particular, it is a birational map f : M1 99K

M2.

We claim that π−11 (V ) ⊂ M1 is an open subset with complement of codimension at least two.

On one hand, from the construction of V we observe that the complement of V in M stX1,σ1

(v1)

has codimension at least two. Together with the fact that π1 : π−11 (M st

X1,σ1(v1))→M st

X1,σ1(v1) is

an isomorphism, we conclude that the complement of π−11 (V ) in π−1

1 (M stX1,σ1

(v1)) also has codi-mension at least two. On the other hand, since V contains an open subset of the singular locusSym2MX1,σ1(v1,p), we obtain that π−1

1 (V ) contains an open subset of the unique exceptional

divisor. Therefore π−11 (V ) has a complement of codimension two in M1.

Now we can apply [GHJ03, Proposition 21.6] to the birational map f : M1 99K M2, and conclude

that π−12 (f(V )) = f(π−1

1 (V )) ⊂ M2 has complement of codimension at least two. This furtherimplies f(V ) ⊂ MX2,σ2(v2) also has complement of codimension at least two. Therefore f(U),as the intersection of f(V ) with the open stratum M st

X2,σ2(v2), has complement of codimension

at least two as well.

2.3. Mukai morphisms are isomorphisms. The Mukai morphism plays an essential role in[BM14a]. A classical theorem [BM14a, Theorem 3.6], originally proved in [Muk87, Yos01], showsthat for a smooth moduli space Mσ(v) of stable objects on a K3 surface X with v2 > 0, theMukai morphism induced by a (quasi-)universal family is in fact a Hodge isometry between theorthogonal complement of v in the total cohomology v⊥,tr ⊂ H∗(X,Z) and H2(Mσ(v),Z). Byrestricting on the algebraic components on both sides, we get an isometry between the orthogonalcomplement of v in the algebraic cohomology v⊥ ⊂ H∗alg(X,Z) and NS(Mσ(v)).

Perego and Rapagnetta generalised this classical result to generic Gieseker moduli spaces ofO’Grady type in [PR13, Theorem 1.7]. Here we will follow their approach to generalise thesame result further to generic Bridgeland moduli spaces of O’Grady type. This is the contentof the following

Theorem 2.7. Let v be a Mukai vector of O’Grady type and σ ∈ Stab†(X) be a generic stabilitycondition with respect to v. Let M = Mσ(v) be the moduli space of σ-semistable objects of class

v on X and π : M →M be its symplectic resolution. Then we have

(1) The pullback map π∗ : H2(M,Z) → H2(M,Z) is injective and compatible with the(mixed) Hodge structures. In particular, the Hodge structure on H2(M,Z) is pure ofweight two and the restriction of the Beauville-Bogomolov quadratic form q(−,−) on

H2(M,Z) defines a quadratic form q(−,−) on H2(M,Z);

(2) There exists a well-defined Mukai morphism θtrσ : v⊥,tr → H2(M,Z) induced by the

(quasi-)universal family over M stσ (v), which is a Hodge isometry.

The proof of Theorem 2.7 is postponed to the next section. We continue our discussion withthe following interesting consequence, which will be very important later.

Corollary 2.8. Under the assumptions of the Theorem 2.7, we have

(1) Lefschetz (1, 1) theorem holds for M . That is, NS(M) = H1,1(M,Z);


(2) The restriction of the pullback map π on NS(X) is an injective map π∗ : NS(M) →NS(M), which is compatible with the Beauville-Bogomolov pairings q(−,−) and q(−,−);

(3) The restriction of the Mukai morphism θtrσ on the algebraic Mukai lattice is an isometry

θσ : v⊥ → NS(M). In particular, q(−,−) is a non-degenerate pairing on NS(M) withsignature (1, ρ(X)).

Proof. For simplicity, we still denote Mσ(v) by M . By Theorem 2.7, the Hodge structure onH2(M,Z) is pure of weight two, and π∗ preserves the Hodge structure. Therefore, for any class

α ∈ H1,1(M,Z), we have π∗α ∈ H1,1(M,Z) and hence π∗α = c1(L) for some line bundle L

on M . By O’Grady’s construction of the symplectic resolution M in [O’G99], we know that ageneric fibre of π within the exceptional divisor is a smooth rational curve. Let C be such a

rational curve, then L ·C = π∗α · [C] = α · π∗[C] = 0, which implies that the restriction of L onC is trivial. Since M is normal by [BM14a, Theorem 3.10] (or originally [KLS06, Theorem 4.4]),

we must have L = π∗L for some line bundle L on M . Therefore we have α = c1(L) ∈ NS(M)and the Lefschetz (1, 1) theorem is true for moduli spaces M of O’Grady type.

By taking the (1, 1) components on both sides of the map π∗ between the second cohomologygroups, we get the map between the Neron-Severi lattices. Similarly, we can take the (1, 1)components on both sides of the Hodge isometry θtr

σ to get the desired isometry θσ.

Remark 2.9. We briefly describe how the map θtrσ will be constructed in the proof of Theorem

2.7. Indeed, we will see that it is the unique lift of ΦE along i∗ in the diagram

v⊥,trθtrσ //

ΦE ))

H2(Mσ(v),Z)

i∗

H2(M st

σ (v),Q),

where ΦE is the classical Mukai morphism induced by the (quasi-)universal family E on thestable locus M st

σ (v) of the moduli space, and i∗ is the pullback along an open embedding. Themap θσ, which is the restriction of θtr

σ , is sometimes also referred to as the Mukai morphismin the literature. However, we prefer to call it the algebraic Mukai morphism, to distinguish itfrom the Mukai morphism θtr

σ , which includes the (2, 0) and (0, 2) components on both sides.

Remark 2.10. We would also like to point out that, although θtrσ and θσ a priori depend on

the choice of the generic stability condition σ, they in fact only depend on the choice of theopen chamber C ∈ Stab†(X) containing σ. This is because the moduli space is the same for allinterior points of C. Therefore, in [BM14b, BM14a], the Mukai morphism and algebraic Mukaimorphism are sometimes also denoted by θtr

C and θC respectively.

We conclude this section by briefly mentioning various cones of divisors on M = Mσ(v). Theample cone Amp(M), big cone Big(M) and movable cone Mov(M) are all well-defined. Due tothe existence of the symplectic resolution and Corollary 2.8, we can also define the positive coneof M . We will show that the following definition justifies the name.

Definition 2.11. The cone (π∗)−1(Pos(M)) ⊂ NS(M) is called the positive cone of M , and isdenoted by Pos(M).

We can see from the following proposition that the notion is reasonably defined and agrees withour intuition.


Proposition 2.12. The positive cone Pos(M) is one of the two components of α ∈ NS(M) :q(α, α) > 0 and contains the ample cone Amp(M).

Proof. In fact, by Corollary 2.8(3), we know that the cone α ∈ NS(M) : α2 > 0 has twocomponents. Together with the map π∗ in Corollary 2.8(2), we know they are precisely the

restrictions of the two components of α ∈ NS(M) : q(α, α) > 0 to NS(M), one of which is

Pos(M). This proves the first statement.

If there is no ample class then there is nothing to prove for the second statement. Otherwise, take

any ample class α ∈ Amp(M) and note that α := π∗α is nef and big on M . By [KM98, Propo-sition 2.61], this implies

∫Mα10 > 0. Now the Beauville-Fujiki relation [GHJ03, Proposition

23.14] implies q(α, α) > 0. Thus we have α ∈ Pos(M) and hence α ∈ Pos(M).

Remark 2.13. A priori, the ample cone Amp(M) could be empty. However, it is proved in[BM14b, Theorem 1.3] that M always carries ample classes. Therefore, between the two com-ponents of the cone of square positive classes on M , Pos(M) can be simply identified as the onewhich contains Amp(M).

3. Proof of Theorem 2.7

This whole section is devoted to the proof of Theorem 2.7. We start with some lemmas aboutdeformations of twisted K3 surfaces and local existence of relative moduli spaces of twistedsheaves. The proof of Theorem 2.7 will follow after these preparations. As in [BM14b, BM14a],when talking about twisted sheaves, we always assume that we have a fixed B-field lift of theBrauer class, which was introduced in [HS05].

3.1. Deformations of twisted polarised K3 surfaces. In this subsection we study defor-mations of a polarised K3 surface which carry a non-trivial Brauer class with a B-field lift. Ourmain results here are Proposition 3.7 and Proposition 3.9. Roughly speaking, up to changing theB-field by an integral class, we can always deform a twisted polarised K3 surface to an untwistedpolarised K3 surface in the period domain. Moreover, if there is a Mukai vector on the initialK3 surface which remains algebraic under deformations, and the initial polarisation is genericwith respect to this Mukai vector, then the deformation can be made so that the polarisationis generic with respect to the corresponding Mukai vector on each fibre. Moreover, Lemma 3.6shows how to find such an integral class so as to amend a given B-field and make the abovedeformations possible.

We briefly recall the necessary notions required for the following discussion. The (cohomological)Brauer group of a K3 surface X is the torsion part of the cohomology group H2(X,O∗X) in theanalytic (or etale) topology. A twisted K3 surface is a K3 surface X equipped with a Brauerclass α. Using the exponential sequence, we can always find a rational class B ∈ H2(X,Q), suchthat its (0, 2)-component maps to α under the exponential map, i.e. exp(B0,2) = α. We callsuch a rational class B a rational B-field lift of the Brauer class α. Note that the B-field lift ofany given Brauer class α is not unique.

For every α-twisted sheaf E, a twisted Chern character of E, and hence a twisted Mukai vectorof E, is defined in [HS05, Proposition 1.2], which depends on the choice of the B-field lift B ofα. The construction there guarantees that it is a B-twisted algebraic class, i.e. a class in theB-twisted algebraic cohomology group H∗alg(X,B,Z) := (exp(B) · H∗alg(X,Q)) ∩ H∗(X,Z), as

defined in [HS05, Remark 1.3].


We introduce the following notion for simplicity of presentation: assume we have a familyof K3 surfaces X → S, and each fibre Xs over the point s ∈ S is equipped with a B-fieldBs ∈ H2(Xs,Q). If these B-fields form a section of the local system over S with fibres given byH2(Xs,Q), then we say the family of B-fields is locally constant over S.

The following lemma shows the locally trivial extension of some cohomology classes could remainalgebraic on each fibre of a deformation.

Lemma 3.1. Let (X,H) be a polarised K3 surface and B a rational B-field lift of a certainBrauer class on X. Let v ∈ H∗(X,Z) be of the form v = (r, qh + rB, a) for some positiveinteger r, rational number q and integer a, where h = c1(H). Then v ∈ H∗alg(X,B,Z), i.e. v isa B-twisted algebraic class.

Moreover, for any flat deformation of the polarised K3 surface (X,H) with locally constant B-fields extending B on X, v also extends to a locally constant section, such that we get a twistedalgebraic class on each fibre.

Proof. A simple computation shows the component of exp(−B) · v in H2(X,Q) is qh, which isa (1, 1)-class. Hence we conclude that v ∈ H∗alg(X,B,Z) is an integral B-twisted algebraic classon X.

Assume we have a deformation over S, such that for every s ∈ S, the K3 surface Xs comes withan ample line bundle Hs and a B-field Bs, which are both locally constant classes. Then thelocally trivial extension of v over each fibre Xs is given by vs = (r, qhs + rBs, a) ∈ H∗(Xs,Z)where hs = c1(Hs) ∈ H2(Xs,Z). The same computation shows that it is a Bs-twisted algebraicclass.

The above lemma leads to the following definition.

Definition 3.2. A class v ∈ H∗(X,Z) satisfying the assumption of Lemma 3.1 is called adeformable B-twisted Mukai vector on X.

The following lemma justifies the universality of this notion.

Lemma 3.3. Let (X,H) be a polarised K3 surface, B a rational B-field, and v a B-twistedMukai vector with its degree zero component r > 0. If H is generic with respect to v, thenthe moduli space MX,B,H(v) of B-twisted H-semistable sheaves with Mukai vector v is alwaysisomorphic to a moduli space MX,B,H′(v

′) where v′ is deformable, and H ′ is generic with respectto v′.

Proof. This is also classical (see, for instance, the proof of [HL10, Theorem 6.2.5]). We writev = (r, c, a) and c1(H) = h. By Lemma 3.14, we can replace the Mukai vector v by v′ =v ·exp(mh) for any m ∈ Z, without changing the moduli space. Moreover, H is still generic withrespect to v′. Note that v′ is B-twisted, therefore the degree two component of v′ · exp(−B) isa rational (1, 1)-class. A simple calculation shows that this class is c+ rmh− rB. When m 0,c+ rmh− rB is an ample class and lies in the same chamber as h. We fix such an m and writeh′ for the primitive integral class on the ray generated by c + rmh − rB in the ample cone.We denote the corresponding ample line bundle H ′, and write c + rmh − rB = qh′ for someq ∈ Q. Then v′ = (r, qh′+ rB, a′) for some a′ ∈ Z is a deformable Mukai vector on the polarisedK3 surface (X,H ′). The construction guarantees that H ′ lies in the interior of a chamber, andhence is generic.

We recall the following fact from linear algebra, which will be used in the proof of Lemma 3.6.


Lemma 3.4. Let V be a real vector space equipped with a (possibly degenerate) real-valuedsymmetric bilinear pairing, whose signature is (n+, n−, n0) (for the positive definite, negativedefinite, and isotropic parts respectively). Let V ′ be a codimension one linear subspace of Vequipped with induced pairing, with signature (n′+, n

′−, n

′0). Then we have n+ − 1 6 n′+ 6 n+,

n− − 1 6 n′− 6 n−, and n0 − 1 6 n′0 6 n0 + 1.

The same statement holds for rational vector spaces with rational-valued pairings.

Proof. The proof is completely elementary by looking at the symmetric matrix representing thesymmetric bilinear pairing. We leave it to the reader.

Remark 3.5. If the symmetric bilinear pairing is non-degenerate, i.e. n0 = 0, then we will alsowrite its signature as (n+, n−) by abuse of notation.

We are now ready to prove a technical lemma, which will be used to deal with the subtlepotential issues of ampleness caused by (−2)-classes in the deformation, and the genericnessof the polarisations. We point out that, despite of the words appearing in the statement, thislemma has nothing to do with ampleness of h or B being a B-field. We state it in this way justto indicate the situation in which we apply it. The proof of this lemma only contains elementarylattice theoretic arguments.

Lemma 3.6. Let h ∈ H2(X,Z) be an ample class, and B ∈ H2(X,Q) be a B-field. Fix anarbitrary positive integer N0. Then there exists B′ ∈ B + H2(X,Z) such that B′2 < 0, and forevery non-zero class g ∈ SpanQh,B′ ∩H2(X,Z) ∩ h⊥, we have g2 < −N0.

Proof. Without loss of generality, we can assume h to be a primitive class. Otherwise we canreplace h by the primitive class in the ray generated by h, which is still an ample class. Moreover,let n by the smallest positive integer such that nB ∈ H2(X,Z). Note that n is in fact the orderof the Brauer class represented by B.

Since h is primitive, the exact sequence of lattices

0→ Zh→ H2(X,Z)→ H2(X,Z)/Zh→ 0

splits non-canonically. If we choose a lift of H2(X,Z)/Zh in the ambient lattice H2(X,Z), sayM , then H2(X,Z) = Zh ⊕M . We can write nB = αBh + βBmB under this decomposition,where βB is a non-negative integer and mB ∈M is a primitive class. Moreover, it is easy to seethat, for any primitive class pM ∈M , the lattice SpanQh, pM∩H2(X,Z) has an integral basisgiven by h, pM.

We do the same thing for the second time. Since mB is primitive, the exact sequence of lattices

0→ ZmB →M →M/ZmB → 0

splits, and we can choose a lift L of M/ZmB in M and write M = ZmB ⊕ L. Again, for anyprimitive class pL ∈ L, the lattice SpanQmB, pL∩M has an integral basis given by mB, pL.In particular, the divisibility of any integral linear combination γ1mB + γ2pL is the highestcommon factor of γ1 and γ2.

Now we take K := h,mB⊥ ∩ L be the sublattice of H2(X,Z) containing all lattice points inL which are perpendicular to both h and mB under the Poincare pairing. In particular, theseclasses are perpendicular to nB. Note that K ⊂ L has corank at most 2, therefore K ⊂ H2(X,Z)has corank at most 4. Since the Poincare pairing on H2(X,Z) has signature (3, 19) (see e.g.[BHPVdV04, Proposition VIII.3.2]), we can use Lemma 3.4 repeatedly to see that the restrictionof the Poincare pairing to K has negative signature at least 15. That is, K contains a negative


definite sublattice of rank at least 15. Therefore, for any N 0, we can find a primitive integralclass c ∈ K such that c2 < −N .

Now let us consider the rank two lattice ΛN = SpanQh, c+B ∩H2(X,Z). We will show thatfor N sufficiently large, it does not contain any (−2)-classes which are perpendicular to h.

We first try to find an integral basis for ΛN . Note that nc + nB = αBh + (nc + βBmB) wherenc + βBmB ∈ M . Since c ∈ L is primitive, the divisibility kN of the class nc + βBmB is thehighest common factor of n and βB. In particular, 1 6 kN 6 n. We write nc+ βBmB = kNpNwhere pN is a primitive class in M . By the discussion above, this shows that h, pN is anintegral basis for ΛN . Therefore, we just need to show that for each pair of integers (x, y)satisfying h · (xh+ ypN ) = 0, we have (xh+ ypN )2 < −N0 for any given N0.

We give an estimate of the intersection matrix of the lattice ΛN with integral basis h, pN.Since h is a fixed class, h2 is a fixed number. And we have

h · pN = h · 1

kN(nc+ βBmB) =

βBkN

(h ·mB).

Since 1 6 kN 6 n, we see that |h · pN | 6 βB(h ·mB) where the right hand side is independentof N . Moreover

p2N =

1

k2N

(nc+ βBmB)2 =1

k2N

(n2c2 + β2Bm

2B).

Since 1 6 kN 6 n and c2 < −N , we see that when N 0 is large and positive, p2N 0 is large

and negative.

Now assume we have a non-zero vector xh + ypN ∈ ΛN with h · (xh + ypN ) = 0. Then we canassume y 6= 0 since otherwise the vector is zero. Thus we have

(xh+ ypN )2 = ypN · (xh+ ypN ) = y(xh · pN + yp2N )

= y(xh · pN + y(h · pN )2

h2+ y(p2

N −(h · pN )2

h2))

= y2(p2N −

(h · pN )2

h2).

From the above discussion, we see that (h·pN )2

h2 is bounded for all choices of N , while p2N 0

when N 0. Since y2 is a positive integer, the above computation shows that (xh+ypN )2 0,hence is smaller than any number −N0 we started with.

Therefore, in the statement of the proposition, we can simply choose B′ = c+B. Note that thechoice of c makes c ·B = 0, hence we also have B′2 = c2 +B2 0 for a sufficiently large choiceof N .

The following proposition shows how to deform a twisted polarised K3 surface to an untwistedpolarised K3 surface.

Proposition 3.7. Let (X,α) be a twisted projective K3 surface with an ample line bundle H,and B ∈ H2(X,Q) be a B-field lift of α. We denote h = c1(H) and assume that B2 < 0 andthe lattice SpanQh,B ∩H2(X,Z) ∩ h⊥ does not contain any (−2)-classes. Then there existsa family of twisted projective K3 surfaces (X ,H) → S over an integral base S, with ample linebundle Hs and locally constant B-field Bs on the fibre Xs of the family at each point s ∈ S, suchthat at s0 ∈ S, we have (Xs0 , Hs0 , Bs0) = (X,H,B), and at s1 ∈ S, we have Bs1 ∈ H1,1(Xs1 ,Q).In particular, it represents a trivial Brauer class on Xs1.


Proof. We can choose S to be a single point if B is of type (1, 1). Otherwise, we study the perioddomain for the deformation of polarised K3 surfaces. Consider the lattice Λ = h⊥ ⊂ H2(X,Z),where h = c1(H) ∈ H2(X,Z). The associated period domain is:

D = σ ∈ P(Λ⊗ C) : σ2 = 0, σσ > 0.

The period domain of marked polarised K3 surfaces is obtained by removing hyperplanes of theform δ⊥ from D for all (−2)-classes δ. Since Λ has signature (2, 19), D has two connected com-ponents, which can be identified by complex conjugation. We denote (the marking) [H2,0(X)]by σ0 and observe that σ0 ∈ D.

In order to deform X so that the B-field becomes of type (1, 1), we consider Λ′ = h,B⊥ ⊂H2(X,Z) and set

D′ = σ ∈ P(Λ′ ⊗ C) : σ2 = 0, σσ > 0.Notice that D′ is in fact a hyperplane section of D. The condition B2 < 0 guarantees that Λ′ hassignature (2, 18). Therefore, D′ is non-empty and has two components, which are contained inthe two components of D respectively. The assumption also guarantees that D′ is not containedin the hyperplane δ⊥ for any (−2)-class δ ∈ Λ. Therefore any period point in D′ away fromthese hyperplanes represents a deformation of X on which B has type (1, 1). Let S ⊂ D beany integral curve joining such a period point and σ0. It parametrises a family of projective K3surfaces after removing at most locally finitely many closed points.

Next we deal with the issue of the genericness of polarisations. We remind the reader thatfor any (twisted) Mukai vector v on a K3 surface with rank r, its discriminant is defined as∆(v) = v2 + 2r2; see [HL10, Section 3.4] or [Lie07, Definition 3.2.1.1].

Lemma 3.8. Let (X,H) be a polarised K3 surface with a rational B-field B. Let v be a B-twisted algebraic class with its degree zero component r > 0. If there is no class ξ ∈ H1,1(X,Z)

satisfying both ξ ·H = 0 and − r2

4 ∆(v) 6 ξ2 < 0, then H is generic with respect to v.

Proof. The proof is contained in [HL10, Theorem 4.C.3], and so we only point out what changehas to be made to adapt it in the twisted case. Using the notation there, ξ = r ·cB1 (F ′)−r′ ·cB1 (F )is a class in H1,1(X,B,Z). However, we realise that cB1 (F ) − rB is the (1, 1)-component ofexp(−B)v, hence is a class in H1,1(X,Q). Similarly, cB1 (F ′)− r′B is also a class in H1,1(X,Q).We note that a different way to write ξ is ξ = r · (cB1 (F ′) − r′B) − r′ · (cB1 (F ) − rB), hence ξis in fact also an untwisted (1, 1) class. So we only need to check classes in H1,1(X,Z) insteadof H1,1(X,B,Z). On the other hand, we realise that the Bogomolov inequality also holds intwisted case by [Lie07, Proposition 3.2.3.13]. Therefore by the same argument as in the proofof [HL10, Theorem 4.C.3], if H is not v-generic, then there is such a ξ satisfying both ξ ·H = 0

and − r2

4 ∆(v) 6 ξ2 < 0. The proof is finished by contradiction.

The following proposition shows how to keep the genericness of polarisations with respect to alocally constant section of deformable Mukai vectors.

Proposition 3.9. Assume we are in the situation of Proposition 3.7. We also assume thatv = (r, qh + rB, a) is a deformable Mukai vector with v2 > 0, and H is generic with respectto v. Furthermore suppose that for each non-zero class g ∈ SpanQh,B ∩ H2(X,Z) ∩ h⊥

we have g2 < − r2

4 ∆(v). Then, in the family constructed in Proposition 3.7, v extends to alocally constant family of deformable Mukai vectors over S with its value vs at the point s ∈ S.Moreover, for every s ∈ S, Hs is generic with respect to vs.


Proof. By Lemma 3.1, it is clear that v can always be extended to a locally constant family ofdeformable Mukai vectors as long as h and B extend. Therefore, we just need to show that,for a generic period point σ ∈ D′ as constructed in the proof of Proposition 3.7, the class h isgeneric with respect to the class v on the projective K3 surface corresponding to σ.

We use the criterion in Lemma 3.8 to determine the genericness. That is, in the period domainD constructed in the proof of Lemma 3.1, we need to remove hyperplane sections of the form

ξ⊥ for every class ξ satisfying both ξ ·H = 0 and − r2

4 ∆(v) 6 ξ2 < 0, so that every period pointin the remaining part corresponds to a projective K3 surface on which the ample line bundlecorresponding to the class h is generic with respect to the locally constant extension of v.

By [PR13, Corollary 4.2], we conclude that these hyperplane sections to be removed from arelocally finite in D. However, we also have to show that D′ is not contained in any of thesehyperplanes, so that we can still find period points to make the B-field have type (1, 1) afterremoving these hyperplane sections. In fact, if D′ ⊂ ξ⊥ for some ξ ∈ H2(X,Z), then ξ ∈SpanQh,B ∩ H2(X,Z). By assumption, if such a class satisfies further ξ · h = 0, then g2 <

− r2

4 ∆(v). Hence ξ⊥ is not one of the hyperplane sections to be removed.

3.2. Local existence of relative twisted moduli spaces and compatibility. In this sub-section, we show that for a family of polarised K3 surfaces with locally constant B-fields anddeformable Mukai vectors, we can always construct relative moduli spaces of twisted sheavesin an (etale or analytic) neighbourhood of an arbitrary point. Moreover, this relative modulispace carries a (quasi-)universal family on its smooth locus. However, this construction wouldinvolve a choice of a uniform Cech 2-cocycle representation of the B-fields over that neighbour-hood, hence there is in general no way to glue these (quasi-)universal families together. We willnevertheless show that on any fixed polarised K3 surface, for different choices of the B-field liftof the same Mukai vector, or for different choices of the Cech 2-cocycle representations of thesame B-field, the moduli spaces are always non-canonically isomorphic to one another. As wewill see later, this is already sufficient for the property of Mukai morphisms being isometries tohold or fail simultaneously over the entire base (if connected).

The materials presented in this subsection are probably well-known to experts. However wecouldn’t find any reference in which these results are explicitly written down. Hence we statethese results with full proofs as below. The first result proves the local existence of relativemoduli spaces of semistable sheaves.2

Proposition 3.10. Let (X ,H) → S be a family of projective K3 surface equipped with a lo-cally trivial family of B-fields, denoted by Bs on each fibre Xs. We also write hs = c1(Hs) ∈H2(Xs,Z). Let vs = (r, qhs + rBs, a) ∈ H∗(Xs,Z) be a locally constant family of deformableMukai vectors with v2

s > 0. Then for each point s0 ∈ S, there is an (etale or analytic) openneighbourhood Us0 ⊂ S, over which there is a relative moduli space M, whose fibre Ms at anypoint s ∈ Us0 is the moduli space of Bs-twisted Hs-semistable sheaves of Bs-twisted Mukai vectorvs, with a (quasi-)universal family on the smooth locus of M.

Proof. We first show that, for every fixed s0 ∈ S, over an open neighbourhood of s0, the B-fieldshave a “uniform” Cech 2-cocycle representation. We write p : X → S for the family, and startwith finitely many (etale or analytic) open subsets of the total space X , which cover the fibreXs0 . We label them by Vi for i in a finite index set. The union ∪iVi is an open subset of the

2We thank Kota Yoshioka for informing us that he was able to obtain a global relative twisted moduli spacewithout universal families a long time ago, in the category of algebraic (or analytic) spaces by gluing along anetale (or analytic) cover of the base.


total space X containing the fibre Xs0 . Therefore p(X\∪i Vi) is a closed subset of S missing thepoint s0. We denote S\p(X\∪i Vi) by Us0 , then for every s ∈ Us0 , Vi ∩Xs is a finite (etale oranalytic) cover of the fibre Xs. Using these open covers, we realise that the Cech complex

⊕iΓ(Vi,Z)→ ⊕i,jΓ(Vi ∩ Vj ,Z)→ ⊕i,j,kΓ(Vi ∩ Vj ∩ Vk,Z)→ · · ·

computes the integral cohomology H∗(Xs,Z) for all s ∈ Us0 simultaneously. In fact, its restric-tion on each fibre Xs for s ∈ Us0 is precisely the Cech complex on that fibre. In other words,any Cech 2-cocycle representation of the B-field Bs0 on a single fibre Xs0 using the open coverVi∩Xs can be canonically extended over Us0 , so that the extension simultaneously representsthe B-fields Bs for all s ∈ Bs0 .

From this point on, everything becomes classical. We can write down a moduli functor forfamilies of twisted semistable sheaves on fibres of p−1(Us0) → Us0 . Note that for each suchfamily we have a global Cech 2-cocycle representation of the locally constant B-field. The GITconstruction of the relative moduli space in the untwisted situation (see, for instance, [HL10,Theorem 4.3.7]) applies in the twisted situation and gives a relative moduli space over Us0 , whosefibre over each point s ∈ Us0 is the moduli space of Bs-twisted Hs-semistable sheaves of classvs over Xs. And a (quasi-)universal family exists on the smooth locus of the relative modulispace.

Remark 3.11. For the construction of the associated family of Azumaya algebras and the connec-tion with Simpson’s stability, we refer the reader to [Yos06] and [Sim94]. Note that the relativemoduli space constructed in Proposition 3.10 is in general not smooth over the parameter space.In particular, in the case of relative moduli spaces of O’Grady type, which is the most interestingcase in this paper, the relative moduli space is not smooth over the parameter space.

The following three lemmas handle the compatibility issues caused by change of Cech 2-cocyclerepresentation of a fixed B-field, change of the B-field lift of a fixed Brauer class, and change ofthe Mukai vector by tensoring with multiples of the polarisation.

Lemma 3.12. Let (X,H) be a polarised K3 surface, B be a rational B-field on X, and vbe a B-twisted Mukai vector of O’Grady type. Let B1 and B2 be two different Cech 2-cocyclerepresentations of the same B-field B, and denote the moduli space of B1-twisted (or B2-twisted)H-semistable sheaves with twisted Mukai vector v by MX,B1,H(v) (or MX,B2,H(v)). Then wecan choose an isomorphism f : MX,B1,H(v) → MX,B2,H(v), such that the Mukai morphisms

θtrB1,H and θtr

B2,H are identified. More precisely, the following diagram commutes

v⊥,trθtrB1,H // H2(MX,B1,H(v),Z)

v⊥,trθtrB2,H

// H2(MX,B2,H(v),Z).

f∗ ∼=

OO(3.1)

In other words, the Mukai morphism is independent of the choice of the Cech 2-cocycle rep-resentation of the B-field B. In particular, θtr

B1,H is an isometry if and only if θtrB2,H is an

isometry.

Proof. This is probably well-known, but we nevertheless give a proof. We first show that thereis an equivalence of the abelian categories F : Coh(X,B1) → Coh(X,B2). Without loss ofgenerality, we can assume that the open cover underlying the two Cech 2-cocycle representationsB1 and B2 are the same. Otherwise, we can always find a common refinement, and restriction is


a canonical way to obtain transition functions for the refined open cover from B1 (or B2). Nowwe assume Ui is the open cover for both B1 and B2, then B1 (or B2) can be represented byB1

ijk ∈ Γ(Uijk,Q) (or B2ijk ∈ Γ(Uijk,Q) respectively). Since they represent the same class

in H2(X,Q), the difference B2ijk − B1

ijk ∈ Γ(Uijk,Q) is a coboundary. That is, there exists a

1-cocycle γij ∈ Γ(Uij ,Q), such that B2ijk −B1

ijk = γij + γjk + γki.

With the above preparation, we can construct the functor F in the following way: for every E ∈Coh(X,B1) given by Ei, ϕij, where ϕij ∈ Γ(Uij ,O∗), we define F (E) to be Ei, ϕij ·exp(γij).A direct computation shows that F (E) ∈ Coh(X,B2). Note that this operation is also invertibleand produces F−1. Therefore F : Coh(X,B1) → Coh(X,B2) is an equivalence of categories(in fact in our case both compositions of F and F−1 are the identity functors). It is also easyto see that F preserves the abelian structures on the two categories.

We also observe that the functor F preserves twisted Chern characters. For this we followthe notations and arguments in proof of [HS05, Proposition 1.2]. Note that the B1-twistedChern character of E is defined to be the (untwisted) Chern character of (the C∞ sheaf) EB1 =Ei, ϕ′ij = ϕij · exp(aij) for some C∞-coboundary representation −aij of B1. We realise

that −aij + γij becomes a C∞-coboundary representation of B2. Therefore, the B2-twistedChern character of F (E) can be defined to be the (untwisted) Chern character of (the C∞ sheaf)F (E)B2 = Ei, ϕ′′ij, where ϕ′′ij = ϕij ·exp(γij) ·exp(aij−γij) = ϕij ·exp(aij). Therefore EB1 and

F (E)B2 are in fact the same sheaf, and the twisted Chern characters chB1(E) = chB

2(F (E)).

In other words, the functor F preserves twisted Chern characters.

Since F preserves twisted Chern characters, it also preserves (twisted) Euler characteristicsand (twisted) Hilbert polynomials, and therefore H-stability. It is also straightforward to seethat the construction of F can be carried out in families of B1-twisted sheaves. In particular,F induces a natural transformation between the moduli functors for B1- and B2-twisted H-semistable sheaves of Mukai vector v, and hence induces an isomorphism on the moduli spacesf : MX,B1,H(v) → MX,B2,H(v). Moreover, if E is a (quasi-)universal family of B1-twisted H-

stable sheaves of Mukai vector v on the smooth locus of the moduli space M stX,B1,H(v), then F (E)

is a (quasi-)universal family of B2-twisted H-stable sheaves of Mukai vector v on the smoothlocus of the moduli space M st

X,B2,H(v). The above argument shows that the two families E and

F (E) have the same twisted Chern character. In particular, the cohomological integral functors

ΦE and ΦF (E) given by the two families are the same. More precisely, we have the commutativediagram

v⊥,trΦE // H2(M st

X,B1,H(v),Q)

v⊥,trΦF (E)

// H2(M stX,B2,H(v),Q).

f∗ ∼=

OO(3.2)

By the construction of the Mukai morphism in [PR13, Section 3], we know that ΦE (resp.

ΦF (E)) uniquely determines θtrB1,H (resp. θtr

B2,H), and the commutativity of (3.2) implies the

commutativity of (3.1) (see also [PR13, Lemma 3.11]). Therefore θtrB1,H is an isomorphism of

lattices if and only if θtrB2,H is an isomorphism of lattices, and by [PR13, Lemma 3.12], θtr

B1,H is

an isometry if and only if θtrB2,H is an isometry.

Lemma 3.13. Let (X,H) be a polarised K3 surface, B be a rational B-field on X, and v be aB-twisted Mukai vector of O’Grady type. Pick any c ∈ H2(X,Q) which is a B-field lift of thezero Brauer class. We denote the moduli space of B-twisted (resp. (B+c)-twisted) H-semistable


sheaves with twisted Mukai vector v by MX,B,H(v) (resp. MX,B+c,H(v)). Then we can choose anisomorphism f : MX,B,H(v)→MX,B+c,H(v), such that the Mukai morphisms θtr

B,H and θtrB+c,H

are identified. More precisely, the following diagram commutes

v⊥,trθtrB,H // H2(MX,B,H(v),Z)

v⊥,trθtrB+c,H

// H2(MX,B+c,H(v),Z).

f∗ ∼=

OO(3.3)

In other words, the Mukai morphism is independent of the choice of the B-field lift of the Brauerclass up to a rational (1, 1) class. In particular, θtr

B,H is an isometry if and only if θtrB+c,H is an

isometry.

We point out that the condition c is a B-field lift for the zero Brauer class contains manyinteresting special cases. The cases that we need are c ∈ H2(X,Z) or c ∈ H1,1(X,Z).

Proof. We follow the same idea as in proof of Lemma 3.12. We fix an open cover Ui onwhich both B and c has a Cech 2-cocycle representation, say B = Bijk ∈ Γ(Uijk,Q) andc = cijk ∈ Γ(Uijk,Q). Since c lifts the zero Brauer class, we have exp(c) = 0 ∈ H2(X,O∗). Inother words, we can find a 1-cocycle γij ∈ Γ(Uij ,O∗), such that exp(cijk) = γij · γjk · γki. Wedefine the functor F : Coh(X,B)→ Coh(X,B+ c) as follows: for every E ∈ Coh(X,B) whichcan be represented by E = Ei, ϕij, we define F (E) = Ei, ϕij · γij. It is easy to check thatF (E) ∈ Coh(X,B + c). It is easy to see that F is an equivalence of categories, and preservesthe abelian structure. We also observe that the same contruction can be done in families.

However, similar to the computation in the proof of Lemma 3.12, we observe that chB(E) =chB+c(F (E)). That is, F preserves twisted Chern character and hence preserves the twisted Eu-ler characteristic and H-stability. This implies that F induces an isomorphism of moduli spacesf : MX,B,H(v) → MX,B+c,H(v). Moreover, if E is a (quasi-)universal family over M st

X,B,H(v),

then F (E) is a (quasi-)universal family over M stX,B+c,H(v), with chB(E) = chB+c(F (E)). In

particular, the cohomological integral functors ΦE and ΦF (E) given by the two families are thesame. Hence the following diagram commutes

v⊥,trΦE // H2(M st

X,B,H(v),Q)

v⊥,trΦF (E)

// H2(M stX,B+c,H(v),Q).

f∗ ∼=

OO(3.4)

Using a similar reasoning as in the proof of Lemma 3.12, we conclude that the commutativity of(3.4) implies the commutativity of (3.3). In particular, θtr

B,H is an isometry if and only if θtrB+c,H

is an isometry.

Lemma 3.14. Let (X,H) be a polarised K3 surface, B be a rational B-field on X, and v bea B-twisted Mukai vector of O’Grady type. We write h = c1(H). For any m ∈ Z, there is acanonical isomorphism f : MX,B,H(v)→MX,B,H(v · exp(mh)), such that the following diagram


commutes

v⊥,trθtrB,H,v //

(−)·exp(mh) ∼=

H2(MX,B,H(v),Z)

(v · exp(mh))⊥,trθtrB,H,v·exp(mh)

// H2(MX,B,H(v · exp(mh)),Z).

f∗ ∼=

OO(3.5)

In particular, θtrB,H,v is an isometry if and only if θtr

B,H,v·exp(mh) is an isometry.

Proof. This is classical. We state a proof using the language similar to the above lemmas.We observe that F : Coh(X,B) → Coh(X,B) with the assignment F (E) = E ⊗ H⊗m isan equivalence of abelian categories and is well defined for families. Moreover the functor Fpreserves H-stability. Therefore it induces an isomorphism of moduli spaces f : MX,B,H(v) →MX,B,H(v ·exp(mh)), and takes a (quasi-)universal family E on M st

X,B,H(v) to a (quasi-)universal

family F (E) on M stX,B,H(v · exp(mh)). We can also write down a diagram as follows.

v⊥,trΦE //

(−)·exp(mh) ∼=

H2(M stX,B,H(v),Q)

(v · exp(mh))⊥,trΦF (E)

// H2(M stX,B,H(v · exp(mh)),Q).

f∗ ∼=

OO(3.6)

We just point out that the extra factor exp(mh) in the twisted Mukai vector and the extra factorexp(mh) coming from the twisted Chern character of the (quasi-)universal family cancel out,thanks to the dual operation we have in the Mukai version of cohomological integral functor(see [HL10, Definition 6.1.12], or [Yos01, Section 1.2], or [PR13, Section 3.2]). This is why thecommutativity of the diagram holds. By the same reasoning as in Lemma 3.12, this implies thecommutativity of (3.5).

3.3. Main proof and consequence. We are now ready to state the proof of Thoerem 2.7. Themost difficult part in the proof is to show that the Mukai morphism is an isometry. We outlinethe idea used in this part of proof. The principal idea has already been established in [PR13]where the case of Gieseker moduli spaces is proven. Perego and Rapagnetta proved their result byshowing that one particular moduli space has this property, and all the other Gieseker modulispaces can be related to it by Fourier-Mukai transforms and deformations, under which thisproperty is stable. Our approach for any Bridgeland moduli space comes in two steps. We firstuse a Fourier-Mukai transform constructed in [BM14b, Lemma 7.3] to relate such a Bridgelandmoduli space to a twisted Gieseker moduli space and then use the deformations constructed inPropositions 3.7 and 3.9 to relate the twisted Gieseker moduli space to an untwisted Giesekermoduli space. Since the property of the Mukai morphism being an isometry is stable under bothoperations, and is already true for the untwisted Gieseker moduli space, we conclude it is truefor the Bridgeland moduli space. A complete proof goes as follows.

Proof of Theorem 2.7. We observe that the statement (1) follows from the very general result[PR13, Lemma 3.1]. The pure weight two Hodge structure on H2(M,Z) is described in [PR13,Definition 3.4] and the induced quadratic form is described in [PR13, Definition 3.5].

From now on we focus on statement (2). We first show that the Mukai morphism is alwayswell-defined. Note that using the (quasi-)universal family on the stable locus M st in M , thereis always a well-defined map θst

v : v⊥,tr → H2(M st,Q). It is just the problem whether this mapcan be lifted to H2(M,Z).


If M is the moduli space of B-twisted H-semistable sheaves with Mukai vector v, where B is arational B-field, v is a Mukai vector of O’Grady type, and H is generic with respect to v, thenthe moduli space M admits a GIT construction, and the construction in [PR13, Section 3.2] canbe applied without change. Nevertheless, we point out that the argument there includes a stepwhich requires the local existence of relative moduli spaces, which is guaranteed by Lemma 3.3and Proposition 3.10.

To show the general case, we use [BM14b, Lemma 7.3] and the discussion thereafter. We canalways find a derived equivalence in the form of a Fourier-Mukai tranform Φ : D(X)→ D(Y,B)for a B-twisted projective K3 surface Y , such that the image Φ(σ) of the generic stabilitycondition σ lies in the Gieseker chamber of Stab†(Y,B), hence is represented by an ample linebundle H, which is generic with respect to the Mukai vector Φ(v). In particular, Φ inducesan isomorphism of the moduli spaces Φ : MX,σ(v) → MY,B,H(−Φ(v)). Moreover, every H-semistable object representing a point in MY,B,H(Φ(v)) is locally free, hence Φ(v) has positiverank. We consider the following diagram

v⊥,trΦ∼=

//

θstv

θtrv

))

(Φ(v))⊥,tr

θstΦ(v)

θtrΦ(v)

uu

H2(M stX,σ(v),Q)

Φ∗H2(M st

Y,B,H(Φ(v)),Q)

H2(MX,σ(v),Z)

i∗

OO

Φ∗H2(MY,B,H(Φ(v)),Z)

i∗

OO

By [Yos01, Proposition 2.4]3, we obtain that the upper square commutes, and the first row is anisometry. It is also easy to see the bottom square also commutes, since both vertical maps in itare functorial and both horizontal maps are induced by the isomorphism of the moduli spaces.By the above discussion, there is a lift θtr

Φ(v) for the maps in the right column. Therefore a lift θtrv

for the left column also exists. This shows that the Mukai morphism θtrv is always well-defined

for any Bridgeland moduli space of O’Grady type.

Now we show that θtrv is an isometry. From the above diagram and [PR13, Lemma 3.12], it

suffices to show it only for every twisted Gieseker moduli space MX,B,H(v) of O’Grady type.By Lemma 3.3, we can change v and H if necessary, so that v is deformable. By Lemma 3.6,we can add an integral class to B if necessary, so that B2 < 0, and for every non-zero class

g ∈ SpanQh,B ∩H2(X,Z) ∩ h⊥, we have g2 < min−2,− r2

4 ∆(v), where h = c1(H) and r isthe degree zero component of v. By the above discussion, r is a positive integer. By Lemmas3.12, 3.13, and 3.14, we observe that the above adjustments do not affect the property that θtr

v

is an isometry. That is, it suffices to prove the property of θtrv being an isometry under these

additional assumptions.

The key step here, is to deform such a moduli space of twisted sheaves to an untwisted modulispace. By Proposition 3.7, we can find a family of polarised K3 surfaces (X ,H) → S over anintegral base S, with fibre (Xs, Hs) over any point s ∈ S. Moreover, the family comes equippedwith a locally constant family of B-fields with Bs ∈ H2(Xs,Q) and such that (Xs0 , Hs0 , Bs0) =(X,H,B) and (Xs1 , Hs1 , Bs1) satisfies Bs1 ∈ H1,1(Xs1 ,Q) for some s0, s1 ∈ S. By Lemma 3.1,the Mukai vector v on the fibre over s0 extends to a locally constant section over S, such thatits value vs at every point s ∈ S is still deformable, hence vs ∈ H∗alg(Xs, Bs,Z). By Proposition3.9, we can further assume that Hs is generic with respect to vs for every point s ∈ S.

3Although this proposition is stated for moduli spaces of sheaves, the calculation in its proof is purely coho-mological, which works for moduli spaces of complexes literally.


We now define T := s ∈ S : θtrBs,Hs,v

is an isometry and claim that T is both open and closed

in S. To show it is open, we consider a point t0 ∈ T . By Proposition 3.10, we can find an (etaleor analytic) open subset Ut0 ⊂ S containing t0, such that there is a relative moduli space Mover Ut0 , whose fibre at each s ∈ Ut0 is the moduli space MX,Bs,Hs(v). As observed in [PR13,Proof of Theorem 1.7], the isomorphisms of lattices and integral bilinear forms are both discreteproperties and therefore have to remain constant in families. Since the isometry holds at thepoint t0, it has to hold for every point s ∈ Ut0 , hence Ut0 ⊂ T . Since t0 is an arbitrary pointof T , we conclude that T is open. To show T is closed, we take an arbitrary point t0 ∈ (S\T ).Then, again by Proposition 3.10, a relative moduli space exists over an open neighbourhood Ut0of t0. Since θtr

v fails to be an isometry at t0, it must fail to be an isometry for every point of Ut0 .Hence Ut0 ⊂ (S\T ) and we conclude that T is closed.

Since S is connected, we either have T = S or T = ∅. However, by the construction of S, thereis a point s1 ∈ S at which Bs1 ∈ H1,1(Xs,Q) and hence represents the zero Brauer class. Ithas been proved in [PR13, Theorem 1.7] that the Mukai morphism is an isometry for untwistedmoduli spaces of O’Grady type. By Lemma 3.13, this implies that Mukai morphism θtr

Bs1 ,Hs1is an isometry. Hence we must have T = S. In particular, the Mukai morphism θtr

Bs0 ,Hs0is an

isometry, which is the arbitrary twisted Gieseker moduli space of O’Grady type we started with.This finishes the proof that θtr

σ is an isometry.

Finally, to show that θtrσ preserves Hodge structure we observe that step 4 in [PR13, Proof of

Theorem 1.7] can be applied without change in our situation. And we are done.

Remark 3.15. After writing this proof down, it was pointed out to us that one can use anobservation of Yoshioka to reduce all of the previous arguments to the untwisted case. Indeed,let Φ : D(X)→ D(Y,B) be an equivalence inducing an isomorphismMX,σ(v)→MY,B,H(−Φ(v))as above. Set w0 := Φ((1, 0, 1))∨ and w1 := Φ((0, 0, 1))∨, where ∨ means the dual, and observethat for a general H ′ which is sufficiently close to H, we have Z := MY,−B,H′(w1) ' X whichis an untwisted K3 surface, since 〈w0,w1〉 = −1. Now, if we let F be a universal family andΨ : D(Y,B)→ D(Z) be a twisted Fourier-Mukai transform such that Ψ(Oy) = FZ×y then wecan apply [Yos06, Theorem 1.7 and Remark 1.6] to E(nH ′) for E ∈MY,B,H′(−Φ(v)) and n 0to reduce to the untwisted case.

In fact, the above proof yields the following interesting consequence.

Corollary 3.16. Let v be a Mukai vector of O’Grady type and σ ∈ Stab†(X) be any genericstability condition with respect to v. Then the symplectic resolution of Mσ(v) is deformationequivalent to the irreducible holomorphic symplectic manifold constructed by O’Grady in [O’G99].

Proof. We use the same Fourier-Mukai transform and deformation argument as in the proofof Theorem 2.7. Applying [BM14b, Lemma 7.3] we obtain an isomorphism Φ : MX,σ(v) →MY,B,H(−Φ(v)). In particular, their symplectic resolutions are isomorphic as well. By Proposi-tions 3.7 and 3.9, we can deform the underlying twisted polarised K3 surface to an untwisted K3surface, and the moduli space of twisted semistable sheaves deforms along with the underlyingK3 surface locally near every point. Now we can apply [PR13, Proposition 2.17] and concludethat the fibrewise resolutions of this equisingular family of moduli spaces are also deformationequivalent. Although these families of singular moduli spaces do not necessarily glue into aglobal family due to the relevance of B-field and its Cech 2-cocycle represention, the singularmoduli spaces in different local families over the same twisted K3 surface are always isomorphicdue to Lemmas 3.12 and 3.13. Therefore, they have isomorphic resolutions. Thus, we can con-clude that the symplectic resolution of Mσ(v) is deformation equivalent to that of a Gieseker


moduli space of untwisted sheaves, which, by [PR13, Theorem 1.6], is deformation equivalent tothe holomophic symplectic manifold constructed by O’Grady in [O’G99].

Remark 3.17. We have learned from [KLS06, Theorem 6.2] and [PR13, Theorem 1.6] that theonly deformation type of irreducible holomorphic symplectic manifold that we can obtain byresolving singular Gieseker moduli spaces of semistable sheaves on K3 surfaces is the one con-structed by O’Grady in [O’G99]. Our Corollary 3.16 shows that, even if we enlarge our scope toresolutions of moduli spaces of Bridgeland semistable objects on K3 surfaces, we still only getthe same deformation type.

Remark 3.18. There is another more straightforward proof of Corollary 3.16, which does notneed to go through twisted Gieseker moduli spaces and their deformations. However, it relies ona result which we will prove later, and a very deep theorem of Huybrechts. In fact, by Theorem5.4, we know that Mσ(v) is birationally equivalent to a Gieseker moduli space MH(v) for someample line bundle H which is generic with respect to v. Therefore, their symplectic resolutionsare also birationally equivalent. By [Huy03, Theorem 2.5], these two symplectic resolutions aredeformation equivalent. By [PR13, Theorem 1.6], they are further deformation equivalent to theirreducible holomorphic symplectic manifold constructed by O’Grady in [O’G99]. We remindthe reader that this shorter proof does not result in a circular argument because the proof ofTheorem 5.4 does not use Corollary 3.16.

4. The Local Bayer-Macrı Map

The most essential ingredient in [BM14b, BM14a] is a linearisation map from Stab†(X) toNeron-Severi group of the moduli space for any primitive Mukai vector v ∈ H∗alg(X,Z). In thispaper we refer to it as the Bayer-Macrı map. Note that the construction of Bayer-Macrı mapdepends on the moduli space, therefore one can define a Bayer-Macrı map for each chamber inStab†(X), which we will call a local Bayer-Macrı map. And we are primarily only interestedin the restriction of the map in that chamber (or its closure). These local Bayer-Macrı mapsare already sufficient for the purpose of proving the projectivity of Bridgeland moduli spaces asin [BM14b]. However, in [BM14a], the authors managed to glue the maps defined on variouschambers together and obtained a global Bayer-Macrı map, which reveals nicely the relationbetween wall crossings on the stability manifold and birational geometry of the Bridgelandmoduli spaces.

In this section, we will follow [BM14b] and generalise the definition of the local Bayer-Macrı mapto any non-primitive Mukai vector v of O’Grady type, which relies greatly on the isomorphismθσ proved in Corollary 2.8(3). By using the local Bayer-Macrı map, we will follow the approachin [BM14b] to show the positivity of some determinant line bundles on the singular modulispaces of O’Grady type. We will discuss the global Bayer-Macrı map in a later section.

4.1. Construction of the local Bayer-Macrı map. The local Bayer-Macrı map was firstdefined in [BM14b, Section 3 and 4], and plays a vital role in [BM14b, BM14a]. This mapestablishes a bridge between the two central spaces in question: the stability manifold of the K3surface, and the Neron-Severi group of the moduli space.

There are mainly two different descriptions of the map. The first description, as defined in[BM14b, Section 3], offers the perfect point of view to understand the positivity theorem [BM14b,Theorem 1.1]. While the second description, defined in [BM14b, Section 4], was used more oftenthereafter (for instance in [BM14a, Section 10]). The two descriptions are equivalent by [BM14b,Proposition 4.4].


Here we briefly recall the second description of the local Bayer-Macrı map, which emphasisesthe role of the algebraic Mukai morphism θC from Corollary 2.8, and is more convenient to usefor our purposes. Although it could be described in a more general situation, we only restrictourselves to the case of non-primitive Mukai vectors v of O’Grady type. In short, fix such aMukai vector v and a chamber C ∈ Stab†(X). Then the composition of the following three mapsis called the local Bayer-Macrı map for the chamber C with respect to the Mukai vector v andis denoted by `C .

Stab†(X)Z−→ H∗alg(X,Z)⊗ C I−→ v⊥

θC−→ NS(M).

We briefly describe each map in the above composition. The first map is simply a forgetful map.For any σ = (Z,P) ∈ Stab†(X), with the central charge Z(−) = (Ω,−), we have Z(σ) = Ω ∈H∗alg(X,Z) ⊗ C which only remembers the central charge. An important feature is that Z is a

covering map on an open subset of the target, which was proved in [Bri08, Section 8] (see also[BM14a, Theorem 2.10]). Note that Z does not depend on v or C.

The second map forgets even more. For any Ω ∈ H∗alg(X,Z)⊗C, we have I(Ω) = Im Ω−(Ω,v) . In

particular, when (Ω,v) = −1, this map simply takes the imaginary part of Ω. This map doesnot depend on C either.

Finally, the third map θC is the algebraic Mukai morphism described in Corollary 2.8. Inparticular, it is an isometry. Note that it is the only map among the three which depends onthe choice of the chamber C.

We fix some notations for later convenience (and to keep consistent with the notations in [BM14b,BM14a]). We write wσ := (I Z)(σ) for the image of any σ ∈ Stab†(X) under the compositionof first two maps. The composition of all three maps is denoted by `C = θC I Z. Although`C is defined for the whole Stab†(X), we are mainly interested in its behavior on the chamber Citself (or rather, the closure of C). When the chamber C is clear from the context, for any σ inthe interior of C, we also write `σ := `C(σ), hence θC(wσ) = `σ. And similarly, for any σ0 on theboundary of C, we also write `σ0 := `C(σ0).

4.2. Positivity via the local Bayer-Macrı map. The projectivity of moduli spaces is oneof the main results in [BM14b]. In [BM14b, Corollary 7.5], it is proved that `σ is ample for ageneric σ in the case of primitive Mukai vectors. However, for non-primitive Mukai vectors, theauthors used a rather indirect approach to prove the projectivity, due to the lack of Yoshioka’stheorem [BM14b, Theorem 6.10] in such cases. As a result, it is proved that `σ is ample forσ in a dense subset of C. However, our Theorem 2.7 allows us to mimic the proof of [BM14b,Corollary 7.5] and show the ampleness of `σ for all σ ∈ C.Proposition 4.1. Let v ∈ H∗alg(X,Z) be a Mukai vector of O’Grady type and C ⊂ Stab†(X) anopen chamber with respect to v. Then the image `σ under the local Bayer-Macrı map defined bythe chamber C is ample for all σ ∈ C.

Proof. As in Theorem 2.7, we write π : M → M for the symplectic resolution of M . By[BM14b, Theorem 1.1], we know that `σ is nef on M and therefore its pullback π∗`σ is nef on

M . Moreover, by Corollary 2.8, we have q(π∗`σ) = q(`σ) = w2σ > 0, where the last inequality

follows from [Bri08, Theorem 1.1]. Now by the Beauville-Fujiki relation [GHJ03, Proposition

23.14], we know that the top self-intersection number of π∗`σ on M is positive. By the bigness

criterion [KM98, Proposition 2.61], we see that π∗`σ is big and nef on M . Furthermore, the basepoint free theorem [KM98, Theorem 3.3] tells us that π∗`⊗bσ is globally generated for b 0.


Now we reduce all the above statements from M to M . Since M has rational singularities, we

have H0(M, π∗`⊗bσ ) = H0(M, `⊗bσ ) for any b, i.e. the global sections of π∗`⊗bσ are precisely thoseobtained by pulling back global sections of `⊗bσ along π. Hence we conclude that `⊗bσ is alsoglobally generated.

Finally, by [BM14b, Theorem 1.1] we see that `⊗bσ is positive on all curves in M . But anyglobally generated line bundle that is positive on all curves is ample; see [Huy99, Proposition6.3].

By a similar argument, we can immediately deduce a weaker property for stability conditionson the wall. The following proposition shows that in the O’Grady situation, `σ0 for any σ0 onthe boundary of C is nef, big, and semiample. Recall that a line bundle is semiample if a certainmultiple of it is base point free. In particular, this result shows that the contraction morphismsπ± in [BM14b, Theorem 1.4] are still well-defined in the O’Grady situation and are birationalmorphisms. We will generalise this result to arbitrary Mukai vectors in Proposition 5.2 from acompletely different point of view.

Proposition 4.2. Let v ∈ H∗alg(X,Z) be a Mukai vector of O’Grady type and C ⊂ Stab†(X) anopen chamber with respect to v. For any stability condition σ0 in the boundary of the chamberC, its image `σ0 under the local Bayer-Macrı map defined by the chamber C is big, nef, andsemiample on MC(v).

In particular, `σ0 induces a birational morphism which contracts curves in MC(v) parametrisingS-equivalent objects with respect to the stability condition σ0.

Proof. In fact, the first two paragraphs in the proof of Proposition 4.1 also work here withoutany change, which prove the first statement. The second statement is an immediate consequenceof [BM14b, Theorem 1.1].

5. Classification of Walls: Results

In this section, we study wall crossing on the stability manifold for non-primitive Mukai vec-tors. More precisely, we prove that the criteria in [BM14a, Theorem 5.7] still gives a completeclassification of walls in the O’Grady situation. Moreover, we will prove that [BM14a, Theorem1.1] also holds in the O’Grady situation, which allows us to glue the local Bayer-Macrı mapsand study the birational geometry of the singular moduli spaces via wall crossings. Due to thetechnicality in the proofs, we will only state our results in this section, while leaving all proofsfor next section.

5.1. Hyperbolic lattice associated to a wall. One of the main tools in [BM14a] is a rank twohyperbolic lattice associated to any wall. More precisely, if we fix a Mukai vector v ∈ H∗alg(X,Z)

with v2 > 0, and a wall W of the chamber decomposition with respect to v, then we can defineHW ⊂ H∗alg(X,Z) to be the set of classes

w ∈ HW ⇔ ImZ(w)

Z(v)= 0 for all σ = (Z,P) ∈ W.

By [BM14a, Proposition 5.1], which also works when v is not primitive, the set HW is a primitiverank two hyperbolic lattice. Conversely, given a primitive rank two hyperbolic sublattice H ⊂H∗alg(X,Z) containing v, we can define a potential wall W in Stab†(X) to be a connected

component of the real codimension one submanifold of stability conditions σ = (Z,P) whichsatisfy the condition that Z(H) is contained in a line. Notice that every (potential) wall is


associated to a unique rank two hyperbolic lattice H whereas each lattice may give rise to many(potential) walls. Roughly speaking, whilst crossing a wall W, all the relevant Mukai vectors liein the same hyperbolic lattice HW . This simple observation reduces the analysis of wall-crossingto elementary lattice-theoretic computations.

We also need to recall the names of some special classes in H. A class u ∈ H is called a sphericalclass if u2 = −2, or an isotropic class if u2 = 0. We say a hyperbolic lattice H is an isotropiclattice if H contains at least one non-zero isotropic class. We say a (potential) wall W is anisotropic wall if its associated hyperbolic lattice H is an isotropic lattice.

After establishing the relation between walls and rank two hyperbolic sublattices, we review thenotions of a positive cone and an effective cone. We assume we have a potential wall W andits associated rank two hyperbolic lattice H. The positive cone PH is a cone in H ⊗ R, whichis generated by integral classes u ∈ H, with u2 > 0 and (v,u) > 0. We call any integral classin PH a positive class. Note that the positive cone PH only depends on the choice of H, not onthe choice of the wall associated to H.

In comparison, the effective cone CW depends on the choice of a potential wall. It is a cone in

H⊗ R, which is generated by integral classes u ∈ H, with u2 > −2 and Re Z(u)Z(v) > 0 for a fixed

σ = (Z,P) ∈ W. This notion is well-defined. Indeed, by [BM14a, Proposition 5.5], this conedoes not depend on the choice of σ ∈ W. We call any integral class in CW an effective class.We also point out that for different walls W associated to the same hyperbolic lattice H, theeffective cone CW might differ by spherical classes. However, they all contain the same positivecone PH.

We also need to make the same genericity assumption as in [BM14a, Remark 5.6].

5.2. First classification theorem. We will classify all the walls in the stability manifold fromtwo points of view. Note that various types of walls have been defined in [BM14b, Section 8]and [BM14a, Definition 2.20]. From now on, we always denote a generic stability condition on awall W by σ0, while two generic stability conditions on two different sides of the W by σ+ andσ−. The chambers which contain σ+ and σ− are denoted by C+ and C− respectively. Moreover,we always assume that σ+ and σ− are sufficiently close to the wall, whose precise meaning iscontained in [BM14a, Proof of Proposition 5.1].

We first classify walls according to whether there exists any σ0-stable objects of class v. We saya (potential) wall associated to the class v is totally semistable, if there is no σ0-stable objectsof class v for a generic stability condition σ0 ∈ W. Otherwise, W is not totally semistable. Inother words, the (potential) wall W is totally semistable if and only Mσ+(v) and Mσ−(v) arecompletely disjoint from each other, andW is not totally semistable if and only if they contain acommon open dense subset Mσ0(v). There is a third way to describe these walls, which revealsmore on the reason why the name is obtained. The wall W is totally semistable if every stableσ+-stable object becomes strictly σ0-semistable, which is equivalent to the condition that everystable σ−-stable object is strictly σ0-semistable. Otherwise W is not totally semistable.

We are ready to state the following numerical criterion for a wall to be totally semistable for anarbitrary class v ∈ H with v2 > 0, regardless of whether it is primitive or has O’Grady type.The theorem generalises the first half of [BM14a, Theorem 5.7].

Theorem 5.1. Let v ∈ H be a positive class with v2 > 0, and W be a potential wall for v.Then W is a totally semistable wall for v if and only if either of the following two conditionsholds


(TS1) there exists an effective spherical class s ∈ H with (s,v) < 0;

(TS2) there exists an isotropic class w ∈ H with (w,v) = 1.

In particular, if v is a non-primitive class, then (TS2) cannot happen and (TS1) is the onlypossibility for a totally semistable wall. As an application of the proof of this theorem, we alsoobtain the bigness of `σ0 for an arbitrary class v, which generalises [BM14b, Theorem 1.4(a)]and Proposition 4.2.

Proposition 5.2. Let v ∈ H be a positive class with v2 > 0, and W be a potential wall for v.For a generic σ0 ∈ W, its image `C±(σ0) under the local Bayer-Macrı map with respect to thechamber C± induces a birational morphism

π± : Mσ±(v)→M±

which contracts curves in Mσ±(v) parametrising S-equivalent objects under the stability condition

σ0, where M± is the image of π± in Mσ0(v).

5.3. Second classification theorem. The second classification result relies on Proposition 5.2(or Proposition 4.2). Since π+ : Mσ+(v) → M+ is a birational morphism, it could be one ofthree possible types: a divisorial contraction, a small contraction, or an isomorphism. In fact, inthe next theorem, we will see that π+ and π− always correspond to the same contraction typebecause the numerical criteria do not tell the difference of two sides of W. So it doesn’t matterwhether the definition is made for π+ or π−.

In the first two cases, since π+ and π− induce actual contractions, the two moduli spaces Mσ+(v)and Mσ−(v) are not the same and hence W is a genuine wall which we call a divisorial wall anda flopping wall respectively; the reason for the name will be explained in Section 6. If π+ andπ− are both isomorphisms, we will see that Mσ+(v) and Mσ−(v) are isomorphic. By the firstclassification Theorem 5.1, they could be either disjoint from each other, in which case we callW a fake wall, or identical with each other, in which case we say that W is not a wall. We referthe reader to [BM14b, Section 8] and [BM14a, Definition 2.20] for the names of these walls.

With these notions at hand, we can now state the second classification result which provides anumerical criterion for each of the three types of contraction discussed above. This generalisesthe second half of [BM14a, Theorem 5.7], but we can only prove it in the O’Grady situation:

Theorem 5.3. Let v ∈ H be a positive class with v = 2vp, where v2p = 2, and W be a potential

wall for v. Then W can be one of the following three types:

• W is a wall inducing a divisorial contraction if and only if at least one of the followingtwo conditions holds:

(BN) there exists a spherical class s with (s,v) = 0, or

(LGU) there exists an isotropic class w with (w,v) = 2.Moreover, the condition (LGU) necessarily implies the condition (BN).

• Otherwise, W is a wall inducing a small contraction if and only if the following conditionholds:

(SC) there exists a spherical class s with 0 < (s,v) 6 v2

2 = 4.

• In all other cases, W is either a totally semistable wall, or not a wall.

We discuss briefly the differences between [BM14a, Theorem 5.7] in the primitive case and ourclassification in the O’Grady setting. The main difference shows up in the condition requiredfor a divisorial wall. On the one hand, the non-primitivity of our Mukai vectors excludes the


possibility of having a Hilbert-Chow wall. On the other hand, we will prove in Lemma 6.4 that(LGU) always implies (BN) for moduli spaces of O’Grady type. Therefore, (BN) is the uniquecondition that is required to characterise a divisorial wall. In other words, every divisorialwall is a wall of Brill-Noether type, while it is also a wall of Li-Gieseker-Uhlenbeck type ifboth (BN) and (LGU) are satisfied at the same time. However, there is a subtle differencebetween the cases (BN) and (LGU) which can be seen by looking at the HN filtration of thegeneric semistable object along the divisor; see Remark 6.5 for more details. Another differencebetween our Theorem 5.3 and [BM14a, Theorem 5.7] is that there are no flopping walls arisingfrom a decomposition v = a + b into positive classes a and b. Indeed, this case cannot happenbecause the O’Grady spaces have such small dimension; see Lemma 6.13 for more details.

The above classification shows us how a moduli space could change when the stability conditionapproaches a wall. Although the contraction induced by a wall can be arbitrarily misbehaved,we can always find a birational map, which identifies open subsets in the moduli spaces Mσ+(v)and Mσ−(v) with complements of codimension at least two. By composing these birationalmaps, we can obtain the same conclusion for any pair of generic stability conditions. This is thecontent of the following result, which generalises [BM14a, Theorem 1.1].

Theorem 5.4. Let v ∈ H be a positive class with v = 2vp, where v2p = 2, andW a potential wall

for v. Then there exists a derived (anti-)autoequivalence Φ of D(X), which induces a birationalmap Φ∗ : Mσ+(v) 99KMσ−(v), such that

• Φ∗ is stratum preserving, and

• one can find open subsets U± ⊂Mσ±(v), with complements of codimension at least two,and Φ∗ : U+ → U− an isomorphism, i.e. Φ∗ is an isomorphism in codimension one.

In particular, Φ∗ identifies the Neron-Severi lattices NS(Mσ+(v)) ∼= NS(Mσ−(v)), as well as themovable cones Mov(Mσ+(v)) ∼= Mov(Mσ−(v)) and big cones Big(Mσ+(v)) ∼= Big(Mσ−(v)).

More generally, the same result is true for any two generic stability conditions σ, τ ∈ Stab†(X).

Remark 5.5. Let us take this opportunity to clarify why an autoequivalence Φ of D(X), whichsends every E ∈ U+ ⊂ Mσ+(v) to an object Φ(E) ∈ U− ⊂ Mσ−(v), induces a birationalmap Φ∗ defined as a morphism on U+. If we let Mσ±(v) denote the moduli stacks of σ±-semistable objects then, by the GIT construction in [BM14b], we have classifying morphismsf± : Mσ±(v) → Mσ±(v) together with the fact that the moduli spaces Mσ±(v) universallycorepresent the moduli stacks Mσ±(v). Now, the autoequivalence Φ defines a morphism froman open substack of Mσ+(v) to an open substack of Mσ−(v), which is in fact an isomorphism(because the functor Φ has an inverse). Note that these two open substacks are in fact preimagesof open subschemes of the corresponding moduli spaces (along f+ and f−); the reason is that ifΦ(E) is σ−-semistable, then every object in the same σ+-S-equivalence class of E is also mappedto a σ−-semistable object. Finally, the isomorphism between the two open substacks descends toan isomorphism between the two open subschemes by the universal corepresentability of Mσ±(v).In particular, if U± ⊂ Mσ±(v) is an open subset then U± universally corepresents its preimage

f−1± (U±) since universal corepresentability is preserved under arbitrary base change; see [HL10,

Definition 2.2.1 & Theorem 4.3.4] for more details.

Remark 5.6. This theorem suggests that we can use the notation NS(M(v)) for the Neron-Severi lattice of the moduli space without specifying the generic stability condition (or rather itschamber). Similarly, we will also use the notations Mov(M(v)) and Big(M(v)) for the movablecone and big cone of the moduli space associated to any chamber in Stab†(X).


The fact that all generic moduli spaces have canonically identified Neron-Severi groups showsthat, all chamberwise local Bayer-Macrı maps have the same target and hence could possibly beglued together into a global Bayer-Macrı map, which will reveal the power of wall crossings inthe study of birational geometry of the moduli spaces; see Section 7.

6. Classification of Walls: Proofs

In this section, we prove all the results which were stated in Section 5. First, we fix notation.Throughout, we set v = mvp ∈ H∗(X,Z) to be a Mukai vector, for some positive integer m, andvp a primitive vector with v2

p > 0. Although we will mainly focus on moduli spaces of O’Grady

type, where m = 2 and v2p = 2, some of our results are actually true for arbitrary v. Let

H ⊂ H∗(X,Z) be a primitive hyperbolic rank two sublattice containing v, and W ⊂ Stab†(X)a potential wall associated to H. A generic stability condition on W is denoted by σ0, whilegeneric stability conditions on the two sides of the wall are denoted by σ+ and σ−. The phasefunctions of the central charges of σ+, σ0, σ− are denoted by φ+, φ0, φ− respectively. We alsouse π± : Mσ±(v)→M± for the morphism induced by `σ0 on the generic moduli spaces Mσ±(v).Finally, we point out that, in this section, for simplicity of notations, any morphism (or birationalmap) between two moduli spaces induced by a derived (anti-)autoequivalence Φ of D(X) is stilldenoted by Φ, by abuse of notation.

We should point out that a large portion of our proofs are, in fact, already contained in [BM14a].Therefore, to avoid repetition we will reference the results and proofs of [BM14a] freely and onlypoint out where the differences are and what additional arguments (if any) need to be added.In particular, our focus will be on explaining the extra effort required to deal with the problemscaused by the presence of the singular locus. Moreover, all results that we prove for σ+ alsohold for σ− with identical proof, which we will not explicitly mention in every statement.

6.1. Totally semistable walls. In this subsection we prove the criterion stated in Theorem 5.1for a wall to be totally semistable. We will also show that π± are always birational morphisms.We prove these results for arbitrary Mukai vectors v with v2 > 0. The notion of the minimalclass in a GH-orbit will be frequently used; for its definition, we refer the reader to [BM14a,Proposition and Definition 6.6].

We start by proving some lemmas. The first two deal with non-minimal and minimal classes re-spectively, whilst the third one establishes a bridge between these two kinds of classes when theylie in the same GH-orbit. This will be used later to deduce results for non-minimal classes fromthe existing results for minimal classes. After that, we will prove Theorem 5.1 and Proposition5.2.

Lemma 6.1. If v is non-minimal, then there is no σ0-stable objects of class v, which means Wis a totally semistable wall.

Proof. This is the first statement in [BM14a, Proposition 6.8]. Note that the proof there worksfor both primitive and non-primitive classes.

Lemma 6.2. Let v be a minimal, non-primitive class. Then there exist σ0-stable objects ofclass v.

Proof. We write v = mvp where m > 1 and vp is a minimal primitive class. There are two casesto consider.


If there is no isotropic class w with (w,vp) = 1, then by [BM14a, Theorem 5.7], there existσ0-stable objects of class vp. Therefore by [BM14a, Lemma 2.16], there exist σ0-stable objectsof class v.

If instead there is an isotropic class w with (w,vp) = 1, then by [BM14a, Proposition 8.2] andthe discussion above it (see also [LQ14, Lo12]), `σ0 induces a Li-Gieseker-Uhlenbeck morphism.Since (w,v) = m > 1, a generic stable object in Mσ±(v) corresponds to a locally free stablesheaf in a Gieseker moduli space, hence remains σ0-stable.

Lemma 6.3. Let v be any class not satisfying (TS2), and v0 be the minimal class in the GH-orbit of v. Then there exists a derived autoequivalence Φ+ of D(X) defined as a composition ofspherical twists, such that for every σ0-stable object E of class v0, Φ+(E) is a σ+-stable objectof class v. Moreover, for any other σ0-stable object E′ of class v0, Φ+(E) and Φ+(E′) are notS-equivalent with respect to σ0.

Proof. By [BM14a, Theorem 5.7] (in primitive case) or Lemma 6.2 (in non-primitive case), therealways exist σ0-stable objects of class v0. The first statement is contained in [BM14a, Proposition6.8], and the second statement is contained in the proof of [BM14a, Corollary 7.3].

Now we are ready to prove the criterion for totally semistable walls, and show that the contrac-tion induced by `σ0 on Mσ±(v) are always birational for an arbitrary Mukai vector v, whichgeneralises Proposition 4.2.

Proof of Theorem 5.1. If v is a primitive class, the statement is proved in [BM14a, Theorem5.7]. Now we assume v is non-primitive. Note that in this case (TS2) cannot happen, hence weonly need to prove that (TS1) is both sufficient and necessary for W to be a totally semistablewall. The sufficiency is the content of Lemma 6.1, and Lemma 6.2 proves the necessity bycontradiction.

Proof of Proposition 5.2. If v is a primitive class, the statement is [BM14b, Theorem 1.4(a)].Now we assume v is non-primitive. Note that by [BM14b, Theorem 1.1], `σ is nef on Mσ+(v),and contracts curves which generically parametrise S-equivalent objects with respect to σ0; seeProposition 4.2. Therefore, it suffices to show that there exists a dense open subset U ⊂Mσ0(v)in which no curve is contracted by `σ0 . If v is minimal, we simply take U to be the opensubset of σ0-stable objects. By Lemma 6.2, U is non-empty and hence dense in Mσ0(v). If vis non-minimal, let v0 be the corresponding minimal class with an open subset U0 ⊂ Mσ+(v0)of σ0-stable objects. Then, by Lemma 6.3, U = Φ(U0) is non-empty and does not contain anycurve which can be contracted by `σ0 .

By virtue of Proposition 5.2, the contractions π± : Mσ±(v)→M± induced by `σ0 are birationalmorphisms, and we will deal with three mutally exclusive cases: a divisorial contraction, asmall contraction, or no contraction at all, i.e. an isomorphism. We will prove the numericalcriterion for each case, and show that Mσ+(v) and Mσ−(v) are always birational and isomorphicin codimension one.

We also point out that from now on, we always restrict ourselves to the O’Grady situation. Thatis, v = 2vp where vp is a primitive Mukai vector with v2

p = 2. We do this because many of thefollowing proofs will rely heavily on the existence of a symplectic resolution.

6.2. Walls inducing divisorial contractions. In this subsection we prove Theorems 5.3 and5.4 in the case that `σ0 induces a divisorial contraction. We remind readers that the singular


locus in the O’Grady moduli spaces has codimension two. Therefore, the contraction of adivisor is a phenomenon which happens in the smooth locus. This is the reason why most ofthe arguments in [BM14a] still apply in the O’Grady situation.

This subsection consists mainly of a series of lemmas; each of which will become an integralpart in the proof of Theorems 5.3 and 5.4. At the end of this subsection, we summarise all theresults with Proposition 6.12. We start with the following special feature of classes of O’Gradytype, which will be used later to simplify a few proofs.

Lemma 6.4. Let v be any class of O’Grady type. If (LGU) holds for v, then (BN) also holdsfor v. Moreover, v is a minimal class, and we can choose the class w in (LGU) so thatMσ0(w) = M st

σ0(w).

Proof. By (LGU) we have (w,vp) = 1 which implies w is a primitive class. Using the notation in[BM14a, Lemma 8.1], we have either (w0,vp) = 1 or (w1,vp) = 1. Suppose (w0,vp) = 1. Thenwe have (vp−w0)2 = v2

p−2 = 0 which can only happen when vp−w0 is a multiple of w0 or w1.

If it is a multiple of w0, then so is vp, which contradicts v2p = 2. Thus, we have vp −w0 = kw1

for some non-negative integer k and hence 2 = v2p = (w0 + kw1)2 = 2k(w0,w1) which implies

k = 1 and (w0,w1) = 1. Swapping subscripts shows that starting from (w1,vp) = 1 also yieldsvp = w0 + w1 and (w0,w1) = 1. Therefore, we always have (vp,w0) = (vp,w1) = 1.

Now, the unique effective spherical class is given, up to sign, by s = w0 − w1. In particular,(s,v) = 2(s,vp) = 0 and we see that (BN) must also hold. We also have M st

σ0(w0) = Mσ0(w0)

and M stσ0

(w1) = ∅, where w1 = w0 + (s,w0)s, and so we can choose the class w in (LGU) to bew0; see [BM14a, Proposition 6.3 and Lemma 8.1] for more details.

Remark 6.5. In the case when (LGU), and hence (BN), is satisfied, the HN filtration along thedivisor does not quite behave like the normal BN case. Indeed, if we look at the arguments of[BM14a, proof of Theorem 1.1, p.40] then we see that the divisor of semistable objects is stillthe BN-divisor, but the HN filtration of the generic semistable object does not have the normalBN-form, i.e. it does not contain a spherical factor.

Lemma 6.6. Let v be a minimal class of O’Grady type. If (BN) does not hold, then the set ofσ0-stable objects in Mσ+(v) has complement of codimension at least two.

Proof. If H does not contain any isotropic vector, we follow the proof of [BM14a, Lemma 7.2].Otherwise, by Lemma 6.4, (LGU) does not hold either. Moreover there is no isotropic class wwith (w,v) = 1 since v is non-primitive. Therefore, we can follow steps 1 and 2 of the proofof [BM14a, Proposition 8.6] to get the conclusion. The only change that we have to make inboth proofs is that the application of their Theorem 3.8 has to be replaced by our Proposition2.3.

Lemma 6.7. Let v be a non-minimal class of O’Grady type. If (BN) does not hold, then thereis an open subset of Mσ+(v) with complement of codimension at least two, on which no curve iscontracted by `σ0.

Proof. We write v0 for the minimal class in the GH-orbit containing v, then (BN) does nothold for v0. By Lemma 6.6, there exists an open subset U0 ⊂ Mσ+(v0) with complement ofcodimension at least two, which parametrises objects remaining stable under σ0. By Lemma6.3, there is a derived autoequivalence Φ+ of D(X) which induces an injective morphism Φ+ :U0 →M st

σ+(v), and the image Φ+(U0) does not contain any curve that generically parametrises

S-equivalent objects under σ0. Therefore, no curve in Φ+(U0) will be contracted by `σ0 . Itremains to show Φ+(U0) has complement of codimension two in Mσ+(v).


Since v0 is a minimal class and (LGU) does not hold (by Lemma 6.4), Theorem 5.1 tells usthat W is not a totally semistable wall for v0,p, i.e. the effective primitive class proportionalto v0. By Lemma 6.3, the same derived auto-equivalence Φ+ takes every σ0-stable object ofclass v0,p to a σ+-stable object of class vp, i.e. the effective primitive class proportional to v.Now by Lemma 2.5, Φ+ : Mσ+(v0) 99K Mσ+(v) is a stratum preserving birational map and sowe can apply Proposition 2.6 to conclude that Φ+(U0) has complement of codimension two inMσ+(v).

Lemma 6.8. Let v be any class of O’Grady type and assume that (BN) holds for v while (LGU)does not hold for v. Then π+ is a divisorial contraction of Brill-Noether type.

Proof. We first consider the case that H contains no isotropic vectors. The divisor contractedby π+ is constructed in [BM14a, Lemma 7.4] when v is a minimal class, and in [BM14a, Lemma7.5] when v is a non-minimal class. Next, we consider the case that H is isotropic. By [BM14a,Proposition 6.3], there is only one effective spherical class in H, hence v is necessarily minimal.The divisor contracted by π+ is constructed in [BM14a, Lemma 8.8], which also shows thecontraction has Brill-Noether type.

Lemma 6.9. Let v be any class of O’Grady type and assume that (BN) holds for v while (LGU)does not hold. Then there is a derived (anti-)autoequivalence Φ inducing a stratum preservingbirational map Φ : Mσ+(v) 99KMσ−(v), which is an isomorphism in codimension one.

Proof. By Lemma 6.8 we know that π+ is a contraction of Brill-Noether type under the givenassumption. If v is a minimal class, then we can follow the proof of [BM14a, Theorem 1.1(b)]in the Brill-Noether case and obtain a spherical twist Φ of D(X), inducing a birational mapΦ : Mσ+(v) 99KMσ−(v), which is an isomorphism in codimension one. It remains to show thatit is stratum preserving. Note that under the given assumption, for the primitive class vp (whichis the primitive part of v), the wall W also induces a Brill-Noether divisorial contraction and isnot totally semistable by [BM14a, Theorem 5.7]. We apply the same part of proof in [BM14a,Theorem 1.1(b)] to conclude that the restriction of Φ on the moduli spaces with primitive classesΦp : Mσ+(vp) 99K Mσ−(vp) is also a birational map. By Lemma 2.5, we conclude that Φ is astratum preserving birational map between moduli spaces with non-primitive classes.

Now we consider the case that v is not a minimal class. We still write v0 for the minimalclass in the GH-orbit of v. The above construction gives Φ0 : Mσ+(v0) 99K Mσ−(v0). By[BM14a, Lemma 7.5], there is a composition of spherical twists, say Φ+, inducing a birationalmap Φ+ : Mσ+(v0) 99K Mσ+(v). Moreover, since W is not a totally semistable wall for theprimitive class v0,p (which is the primitive part of v0), we can apply [BM14a, Proposition 6.8]to see that Φ+ also induces a birational map Φ+,p : Mσ+(v0,p) 99K Mσ+(vp) and Lemma 2.5shows that Φ+ is a stratum preserving birational map. By [BM14a, Lemma 7.5], there existsan open subset U+ ⊂ Mσ+(v0) with complement of codimension at least two, such that Φ+ isan injective morphism on U+ and so by Proposition 2.6, we see that Φ+ is an isomorphism incodimension one.

In the same way we can construct Φ− : Mσ−(v0) 99K Mσ−(v). Then the composition Φ− Φ0 Φ−1

+ : Mσ+(v) 99K Mσ−(v) is a stratum preserving birational map which is isomorphic incodimension one, as desired.

Lemma 6.10. Let v be any class of O’Grady type and assume that (LGU) holds for v. Thenπ+ is a divisorial contraction of Li-Gieseker-Uhlenbeck type.


Proof. By Lemma 6.4, v is a minimal class and we can assume Mσ0(w) = M stσ0

(w) for the classw in (LGU). We can simply follow the proof of [BM14a, Lemma 8.7] to get the divisor contractedby π+, and it is clear from there that the contraction is of Li-Gieseker-Uhlenbeck type.

Lemma 6.11. Let v be any class of O’Grady type and assume (LGU) holds for v. Then thereis a derived (anti-)autoequivalence Φ of D(X) inducing a stratum preserving birational mapΦ : Mσ+(v) 99KMσ−(v), which is an isomorphism in codimension one.

Proof. The proof of this lemma is similar to that of Lemma 6.9. However, in this situation,whileW is a Li-Gieseker-Uhlenbeck type divisorial wall for the class v, it is a Hilbert-Chow typetotally semistable divisorial wall for the class vp. By Lemma 6.4, v and hence vp, are minimalclasses. The proof of [BM14a, Theorem 1.1(b)] in the Li-Gieseker-Uhlenbeck case shows thata derived anti-autoequivalence Φ (which is in fact a spherical twist) induces a birational mapΦ : Mσ+(v) 99K Mσ−(v) which is isomorphic in codimension one. The same proof as in theHilbert-Chow case shows that Φ also induces a birational map Φp : Mσ+(vp) 99K Mσ−(vp).Therefore by Lemma 2.5, Φ is also a stratum preserving birational map.

We conclude the discussion about divisorial walls by summarising all the above lemmas and prov-ing the following proposition, which is nothing but the collection of statements about divisorialcontractions in Theorems 5.3 and 5.4.

Proposition 6.12. The potential wall W associated to a rank two hyperbolic lattice H ⊂H∗alg(X,Z) containing v induces a divisorial contraction if and only if at least one of the followingtwo conditions holds


(LGU) there exists an isotropic class w with (w,v) = 2.

Moreover, the condition (LGU) necessarily implies the condition (BN).

In either case, we can find an (anti-)autoequivalence Φ of D(X), which induces a birational mapΦ : Mσ+(v) 99KMσ−(v). It is stratum preserving and an isomorphism in codimension one.

Proof. Lemma 6.4 shows that (LGU) implies (BN). Therfore it suffices to show W induces adivisorial contraction if and only if (BN) holds. The necessity of (BN) is proved in Lemma 6.6for minimal classes and in Lemma 6.7 for non-minimal classes by contradiction. The sufficiencyof (BN), as well as the type of contraction, are the contents of Lemmas 6.8 and 6.10. Finally,the birational map Φ is constructed in Lemma 6.9 for Brill-Noether contractions and in Lemma6.11 for Li-Gieseker-Uhlenbeck contractions.

6.3. Walls inducing small contractions or no contractions. In this subsection, we proveTheorems 5.3 and 5.4 in the case thatW induces a small contraction or no contraction at all. Wepoint out that most of the results in this subsection work for all effective classes v of divisibilitytwo, because the existence of a symplectic resolution will not be used in the proof.

As before, this subsection consists mainly of a series of lemmas; each of which will becomean integral part in the proof of Theorems 5.3 and 5.4. All the results will be summarised inPropositions 6.21 and 6.22 at the end. These two propositions, together with Proposition 6.12,cover all the cases in Theorems 5.3 and 5.4.

Lemma 6.13. Let M = Mσ(v) be a moduli space of O’Grady type. Then there are no smallcontractions arising from a decomposition of v into the sum a + b of two positive classes.


Proof. Suppose we have such a decomposition for v. Then

8 = v2 = a2 + 2(a,b) + b2 > 2(a,b) =⇒ (a,b) 6 4.

If we suppose further that W does not induce a divisorial contraction, then just as in the proofof [BM14a, Proposition 9.1], we can assume that (a,b) > 2. This leaves (a,b) = 3 or 4 as theonly cases. Next, let us observe that we also have the following equation:

8 = v2 = (v,a + b) = (v,a) + (v,b) = 2(vp,a) + 2(vp,b).

That is, (v,a) and (v,b) must be even integers. In particular, the following inequality

(v,a) = (a + b,a) = a2 + (a,b) > (a,b) > 2

implies that (v,a) = 4; and hence (v,b) = 4 as well. In other words, we are left with two caseswhen 4 = (v,a) = a2 + (a,b):

• If (a,b) = 3 then a2 = 1; contradicting the fact that the lattice is even.

• If (a,b) = 4 then a2 = 0 which implies s := vp − a is a spherical class with (s,v) = 0.Hence, by Proposition 6.12, we must be on a BN-wall which contradicts our assumptionthat W does not induce a divisorial contraction.

Thus, writing v as the sum of two positive classes a and b never gives rise to a flopping wall.

Remark 6.14. Notice that Lemma 6.13 makes our classification of small contractions considerablyeasier than that of [BM14a]. In particular, for a two-divisible Mukai vector, the parallelogramdescribed in [BM14a, Lemma 9.3] always contains an extra interior lattice point vp. Even thoughit is possible to modify the proof of [BM14a, Lemma 9.3] in order to deal with two-divisible Mukaivectors, we will not need it here when v2

p = 2.

Lemma 6.15. Let v be a minimal class of O’Grady type, and assume that W does not inducea divisorial contraction for v. If (SC) does not hold for v, then W is not a wall for v.

Proof. We use the decomposition of the moduli space Mσ±(v) = M stσ±(v) t Sym2Mσ±(vp).

Notice that vp is minimal since v is, and so by Lemma 6.13 and [BM14a, Proposition 9.4], weknow that W is not a wall for vp. In particular, we can identify the strictly semistable lociSym2Mσ+(vp) = Sym2Mσ−(vp). For the stable loci, we observe that the proof of [BM14a,Proposition 9.4] can be applied without change to show that every σ+-stable object of class vis also σ0-stable, and hence σ−-stable. (In fact, the only extra point is that any σ−-unstableobject of class v cannot have two HN-factors of class vp.) The same is true for σ−-stable objects.Therefore, we can also identify the stable loci M st

σ+(v) ∼= M st

σ−(v) and conclude that W is not awall for v.

Lemma 6.16. Let v be a non-minimal class of O’Grady type, and assume that W does notinduce a divisorial contraction for v. If (SC) does not hold for v, then W is a fake wall forv. Moreover, there exists a derived autoequivalence Φ of D(X), which induces an isomorphismΦ : Mσ+(v) ∼= Mσ−(v).

Proof. Let v0 be the minimal class in the same GH-orbit as v. Then as in the proof of [BM14a,Proposition 9.4], we consider the derived autoequivalence Φ+ of D(X) (which is in fact a com-position of spherical twists) constructed in [BM14a, Proposition 6.8].

On the one hand, [BM14a, Proposition 9.4] shows that Φ+ induces an isomorphism Φ+,p :

Mσ+(v0,p)∼−→ Mσ+(vp) on the moduli spaces with primitive classes. Since Φ+ preserves exten-

sions, it also induces an isomorphism Φ+ : Sym2Mσ+(v0,p)∼−→ Sym2Mσ+(vp) on the strictly

semistable loci of the moduli spaces with non-primitive classes. Moreover, since the S-equivalence


relation with respect to σ0 is trivial on Mσ+(v0,p), it is also trivial on Mσ+(vp), and hence on

Sym2Mσ+(vp), which implies that no curve in Sym2Mσ+(vp) is contracted by π+.

On the other hand, Lemma 6.15 ensures that every σ+-stable object of class v0 is also σ0-stableand so by Lemma 6.3, we have an injective morphism Φ+ : M st

σ+(v0) → M st

σ+(v) whose image

does not contain any curve contracted by π+. Now we combine the stable and strictly semistableloci to get an injective morphism Φ+ : Mσ+(v0)→Mσ+(v). Notice that Φ+ is an isomorphism

from Mσ+(v0) to its image in Mσ+(v) because the inverse derived autoequivalence Φ−1+ induces

an inverse of the morphism. Finally, since both moduli spaces are irreducible projective varietiesof the same dimension, Φ+ is in fact an isomorphism between the two moduli spaces. Moreover,by the above discussion, no curve in Mσ+(v) is contracted by π+. By Theorem 5.1, we knowthat W is totally semistable for v and so it must be a fake wall.

Similarly we have a derived autoequivalence Φ− ofD(X) inducing an isomorphism Φ− : Mσ−(v0)∼−→

Mσ−(v). Since W is not a wall for v0, the composition Φ− Φ−1+ : Mσ+(v)

∼−→Mσ−(v) gives thedesired isomorphism between the two moduli spaces.

Lemma 6.17. Let v be a Mukai vector of O’Grady type and assume that W does not induce adivisorial contraction for v. If v (or rather vp) satisfies the following boundary condition:

(BC) there exists a spherical class s ∈ H with 0 < (s,vp) 6v2p

2

then W induces a small contraction for v.

Proof. First let us observe that a similar argument to the one used in the proof of Lemma 6.13shows that no small contractions can come from a decomposition of vp into positive classes. Now,if (BC) holds, then by [BM14a, Proposition 9.1], W induces a (divisorial or small) contractionfor the primitive class vp. This implies W induces a contraction in the strictly semistable locusSym2Mσ+(vp) ⊂Mσ+(v), which is a small contraction considered in Mσ+(v).

Lemma 6.18. Let v be a Mukai vector of O’Grady type. Assume that W does not induce adivisorial contraction for v, and condition (BC) does not hold for v. If there exists an effective

spherical class s with 0 < (s,v) 6 v2

2 , then W induces a small contraction for v.

Proof. We first observe that since (BC) does not hold, we in fact have (s,v) = 2(s,vp) > v2p.

Next, following the notation in [BM14a, Proof of Proposition 9.1], we just need to show that

ext1(S, F ) > ext1(E1, E2). This will ensure that all objects obtained by taking extensions of Sand F do not fall in a single S-equivalence class with respect to σ+. To show the inequality,

we observe that ext1(S, F ) > (s,v − s) = (s,v) + 2 > v2p + 2, while ext1(E1, E2) = v2

p + 2 or

v2p, depending on whether E1 and E2 are isomorphic or not. In either case, the inequality is

satisfied and therefore, the contraction is justified.

Lemma 6.19. Let v be a Mukai vector of O’Grady type. Assume that W does not induce adivisorial contraction for v, and condition (BC) does not hold for v. If there exists a non-

effective spherical class s with 0 < (s,v) 6 v2

2 , then W induces a small contraction for v.

Proof. As in Lemma 6.18, the failure of (BC) implies that (s,v) > v2p. We set t = −s (which is

effective) and follow the third and fourth paragraphs in the proof of [BM14a, Proposition 9.1].There are two steps in the proof which require further justification.

As before, we need to show that all the extensions constructed from a fixed F and T as in[BM14a, Proposition 9.1] cannot lie in the same S-equivalence class with respect to σ+ by a


dimension comparison. The proof of [BM14a, Proposition 9.1] shows that (t,v′) = (s,v) + 2and all extensions constructed there are parametrised by a Grassmannian with dim Gr((s,v) +

1, ext1(T , F )) > dim Gr((s,v) + 1, (s,v) + 2) = (s,v) + 1 > v2p + 1. On the other hand, the

dimension of the S-equivalence class of extensions of σ+-stable objects E1 and E2 of class vp isgiven by ext1(E1, E2)−1 = v2

p+1 or v2p−1, depending on whether E1 and E2 are isomorphic or

not. In either case, we have the desired inequality and the construction in [BM14a, Proposition9.1] yields a proper contraction under the given assumption.

Lemma 6.20. Let v be any class of O’Grady type and assume that W induces a small con-traction for v. Then there is a derived autoequivalence Φ of D(X), which induces a stratumpreserving birational map Φ : Mσ+(v) 99K Mσ−(v), which is an isomorphism in codimensionone.

Proof. When v is a minimal class, we claim that we can simply take Φ to be the identity functoron D(X). Since W induces a small contraction, Lemma 6.6 tells us that Mσ+(v) and Mσ−(v)contain a common open subset M st

σ0(v), with complement of codimension at least two in either

moduli space. Therefore, the identify functor induces a birational map from Mσ+(v) to Mσ−(v)which is an isomorphism in codimension one. Notice that the assumptions imply that W is nota totally semistable wall for the primitive class vp and so the identity functor also induces abirational map between Mσ+(vp) and Mσ−(vp). By Lemma 2.5, we conclude that the birationalmap induced by the identify functor is stratum preserving.

As usual, when v is not a minimal class, we denote the corresponding minimal class by v0. LetΦ0 : Mσ+(v0) 99KMσ−(v0) be the birational map induced by the identity functor on D(X). Bythe discussion above, we know that it is stratum preserving and an isomorphism in codimensionone. Now we take Φ+ to be the composition of spherical twists constructed in [BM14a, Proposi-tion 6.8]. By Lemma 6.3, Φ+ induces a birational map Φ+ : Mσ+(v0) 99KMσ+(v), such that itsrestriction on the open subset M st

σ0(v0) is an injective morphism. Moreover, since W is not to-

tally semistable for v0,p, Φ+ also induces a birational morphism Φ+,p : Mσ+(v0,p) 99KMσ+(vp).Thus, by Lemma 2.5, the birational map induced by Φ+ is stratum preserving. Since M st

σ0(v0)

has a complement of codimension at least two in Mσ+(v0), we conclude that Φ+ is an isomor-phism in codimension one by Proposition 2.6. We can follow the same precedure to get a derivedautoequivalence Φ− which induces a birational map Φ− : Mσ−(v0) 99K Mσ−(v) satisfying both

requirements. Then Φ− Φ0 Φ−1+ is the autoequivalence which induces the desired birational

map.

We conclude the discussion by summarising all the lemmas above with the following two propo-sitions; which verify the collection of statements about small contractions and no contractionsin Theorems 5.3 and 5.4.

Proposition 6.21. The set W is a wall inducing a small contraction if and only if it does notinduce any divisorial contraction, and the following condition holds:


2 .

Moreover, we can find an autoequivalence Φ of D(X), which induces a birational map Φ :Mσ+(v) 99KMσ−(v). It is stratum preserving and an isomorphism in codimension one.

Proof. The necessity of (SC) is proved in Lemma 6.15 for minimal classes and in Lemma 6.16for non-minimal classes by contradiction. The sufficiency is proved in Lemma 6.17 when theextra condition (BC) holds and Lemmas 6.18 and 6.19 when (BC) does not hold and (SC) holds.Finally, the birational map is constructed in Lemma 6.20.


Proposition 6.22. In the case that W does not induce any contractions at all, we can findan autoequivalence Φ of D(X), which induces an isomorphism Φ : Mσ+(v) → Mσ−(v). Inparticular, Φ is just the identity functor when v is a minimal class. In other words, W is a fakewall if v is non-minimal, or is not a wall if v is minimal.

Proof. This is contained in Lemma 6.15 for minimal classes and Lemma 6.16 for non-minimalclasses.

To conclude, we observe that all the cases in Theorems 5.3 and 5.4 have been covered byProposition 6.12 (in the case of divisorial contractions), Proposition 6.21 (in the case of smallcontractions), and Proposition 6.22 (in the case of no contractions at all).

7. The Global Bayer-Macrı Map

In this section we introduce the global Bayer-Macrı map and show how it can be used to studythe birational geometry of the singular moduli spaces. We start by discussing its construction.

7.1. Construction of the global Bayer-Macrı map. We use our Theorems 5.3 and 5.4 toglue the local Bayer-Macrı maps defined on chambers to a global map on Stab†(X). The keystep is to prove a compatibility result between any two local Bayer-Macrı maps `C+ and `C−which are defined on adjacent chambers C+ and C− separated by a wall W. The following is ageneralisation of [BM14a, Lemma 10.1].

Lemma 7.1. The maps `C+ and `C− agree on the wall W. More precisely, we have

• If W induces a divisorial contraction, then the analytic continuations of `C+ and `C−differ by the reflection of NS(Mσ+(v)) (or NS(Mσ−(v))) at the divisor D contracted by`σ0;

• In all other cases, the analytic continuations of `C+ and `C− agree with each other.

Proof. Although the Mukai vector v is not primitive in our situation, a (quasi-)universal familystill exists on the stable locus M st

σ+(v) (and M st

σ+(v)) which has a complement of codimension

two. Therefore, the proof for primitive Mukai vectors in [BM14a, Lemma 10.1] still workswithout any changes. In fact, our situation is even easier since Hilbert-Chow wall do not existany more.

As pointed out in [BM14a], we conclude from this lemma that the moduli spaces Mσ±(v) for thetwo adjacent chambers are isomorphic whenW induces a divisorial contraction or no contraction.If W induces a small contraction, then these moduli spaces differ by a flop; this is why thesewalls are called flopping walls in [BM14b, BM14a].

Remark 7.2. By Theorem 5.4 and Lemma 7.1, the local Bayer-Macrı map `C : C →MC(v) definedon each chamber C ⊂ Stab†(X) can be glued together to give a continuous map on Stab†(X).By Remark 5.6, we can denote the global Bayer-Macrı map by ` : Stab†(X)→ NS(M(v)).

7.2. Birational geometry via the global Bayer-Macrı map. We follow the approach in[BM14a, Section 10] to study the global properties of the Bayer-Macrı map. This map allowsus to prove a precise relationship between the birational geometry of the moduli spaces andwall-crossing in the stability manifold. Before we state the main theorem of the paper, we needthree more lemmas; the first of which partly describes the image of the global Bayer-Macrı map.


Lemma 7.3. The image of the global Bayer-Macrı map is contained in Big(M(v))∩Mov(M(v)).

Proof. Let σ ∈ Stab†(X) be a generic stability condition with respect to v. Then Proposition 4.1shows that `σ is ample on Mσ(v) and is therefore a big and movable class. By Proposition 4.2or Proposition 5.2, `σ0 is also a big and movable class, because it induces a birational morphismπ+ : Mσ+(v)→M+.

The second lemma tells us that each open chamber in NS(M(v)) which represents the amplecone of a certain birational model is either completely contained in the image of `, or has nointersection with the image of ` at all.

Lemma 7.4. For any generic stability condition σ ∈ Stab†(X), the ample cone Amp(Mσ(v))of the moduli space Mσ(v) is contained in the image of the global Bayer-Macrı map `.

Proof. By Proposition 4.1, we know that `σ ∈ Amp(Mσ(v)). Hence there is at least one point inAmp(Mσ(v)), which lies in the image of `. For any other point α ∈ Amp(Mσ(v)) we can appealto Proposition 2.12 to see that α ∈ Pos(Mσ(v)). Hence, by Corollary 2.8, we have α = θσ(a)for some class a ∈ v⊥ with a2 > 0. Now we use the argument in [BM14a, Proof of Theorem 1.2(a)(b)(c)] to conclude that α also lies in the image of `.

Now we state the third and final lemma. If a wall W of a chamber C in Stab†(X) induces acontraction of the moduli space MC(v), then the image `(W) ⊂ NS(M(v)) ofW under the globalBayer-Macrı map ` : Stab†(X) → NS(M(v)) is a wall of the nef cone Nef(MC(v)). However, ifW is a fake wall, then its image under the global Bayer-Macrı map is not a wall in NS(M(v)).This is the content of the following result.

Lemma 7.5. Let σ0 be a generic stability condition on a fake wall W. Then its image lies inthe interior of the nef cone Nef(Mσ+(v)) of the moduli space Mσ+(v). The same statement istrue for Mσ−(v).

Proof. This is just [BM14a, Proposition 10.3]; whose proof works regardless of whether v isprimitive or not.

We can now state the main theorem of the paper, which crystallises the relation between thebirational geometry of singular moduli spaces of O’Grady type and wall crossings in the stabilitymanifold Stab†(X). It is a slight generalisation of [BM14a, Theorem 1.2] in the case of singularmoduli spaces which admit symplectic resolutions.

Theorem 7.6. Let v be a Mukai vector of O’Grady type and σ ∈ Stab†(X) a generic stabilitycondition with respect to v. Then

(1) We have a globally defined continuous Bayer-Macrı map ` : Stab†(X) → NS(Mσ(v)),which is independent of the choice of σ. Moreover, for any generic stability conditionτ ∈ Stab†(X), the moduli space Mτ (v) is the birational model corresponding to `τ .

(2) If C ⊂ Stab†(X) is the open chamber containing σ, then `(C) = Amp(Mσ(v)).

(3) The image of ` is equal to Big(Mσ(v)) ∩ Mov(Mσ(v)). In particular, every K-trivialQ-factorial birational model of Mσ(v) which is isomorphic to Mσ(v) in codimension 1appears as a moduli space Mτ (v) for some generic stability condition τ ∈ Stab†(X).

Proof. All the ingredients of the proof have already been presented above. For (1), the exis-tence of the global Bayer-Macrı map is contained in Remark 7.2. The ampleness statement in


Proposition 4.1 implies that Mτ (v) is the birational model corresponding to `τ for any genericτ .

For (2), we first realise that `(C) has full dimension by Lemma 7.4 and hence must be an opensubset of Amp(Mσ(v)) by Proposition 4.1. Now we know that the image of every non-fake wallof C has to be a boundary component of Amp(Mσ(v)) whereas Lemma 7.5 ensures that theimage of a fake wall of C never separates Amp(Mσ(v)) into two parts. Therefore, the image ofC under ` is the whole ample cone.

For the first claim in (3), one direction of inclusion is in Lemma 7.3. We just need to showthat every open chamber of Big(Mσ(v)) ∩ Mov(Mσ(v)) regarded as the ample cone of somebirational model of Mσ(v) is contained in the image of `. Assume we have two such ample conesadjacent to each other, one of which lies in the image of ` but not the other. Then the wallseparating the two ample cones must be the image `(W) of a wall W in Stab†(X). By Lemma7.5, W cannot be a fake wall since fake walls are not mapped to walls in the movable cone by`. By Lemma 7.1, W cannot be a flopping wall since the image of ` would extend to the otherside of `(W). Hence W must be a divisorial wall. Then we claim that the image of W hasreached the boundary of Mov(Mσ(v)). In fact, since `σ0 induces a divisorial contraction for ageneric σ0 ∈ W, `σ0 has degree zero on each curve contained in the fibres of this contraction,which implies that any line bundle on the other side of `(W) must have negative degree on thesecurves contained in fibres. Therefore, its base locus contains the entire contracted divisor. Thesecond claim follows immediately from the surjectivity.

Remark 7.7. Careful readers might have found that our proof of surjectivity of the Bayer-Macrımap onto the intersection of the big and movable cones is slightly different from that in [BM14a,Proof of Theorem 1.2 (b)]. More precisely, we avoided using the statement that any big andmovable class is strictly positive, which is well-known for irreducible holomorphic symplecticmanifolds, but not for moduli spaces of O’Grady type. However, using [Bri08, Theorem 1.1],our Theorem 7.6 implies that the same statement is still true for singular moduli spaces ofO’Grady type.

7.3. Torelli theorem for singular moduli spaces of O’Grady type. In analogy with[BM14a, Corollary 1.3], we can prove a Torelli-type theorem for the singular moduli spacesof O’Grady type. It gives a Hodge-theoretic criterion for the existence of stratum-preservingbirational maps between these moduli spaces. We use the same notation as in [BM14a], whichwas originally introduced by Mukai: the total cohomology H∗(X,Z) of a K3 surface X carries aweight two Hodge structure which is polarised by the Mukai pairing. We write v⊥,tr ⊂ H∗(X,Z)for the orthogonal complement of v in the total cohomology.

Corollary 7.8. Let X and X ′ be two smooth projective K3 surfaces with v ∈ H∗alg(X,Z) and

v′ ∈ H∗alg(X ′,Z) Mukai vectors of O’Grady type. If σ ∈ Stab†(X) and σ′ ∈ Stab†(X ′) are

generic stability conditions with respect to v and v′ respectively, then the following statementsare equivalent

(a) There is a stratum preserving birational map ϕ : MX,σ(v) 99KMX′,σ′(v′).

(b) The embedding v⊥,tr ⊂ H∗(X,Z) of the integral weight two Hodge structures is isomor-phic to the embedding v′⊥,tr ⊂ H∗(X ′,Z).

(c) There is an (anti-)autoequivalence Φ from D(X) to D(X ′) with Φ∗(v) = v′.

(d) There is an (anti-)autoequivalence Ψ from D(X) to D(X ′) with Ψ∗(v) = v′ which inducesa stratum preserving birational map ψ : MX,σ(v) 99KMX′,σ′(v

′).


Proof. The proof of [BM14a, Corollary 1.3] already contains most of what we need in our situ-ation. We only point out the differences. For (a) ⇒ (b), we observe that MX,σ(vp) ⊂ MX,σ(v)and MX′,σ′(v

′p) ⊂ MX′,σ′(v

′) are the deepest singular strata and so the stratum preserving bi-rational map ϕ in (a) restricts to a birational map Mσ(vp) 99KMσ′(v

′p). Therefore, by [BM14a,

Corollary 1.3], we can deduce (b). For (b)⇒ (c), the proof is already in [BM14a, Corollary 1.3].For (c) ⇒ (d), we can follow the proof of [BM14a, Corollary 1.3] and replace Φ if necessary,so that there exists a generic τ ∈ Stab†(X ′) with Mσ(v) ∼= Mτ (v′). By Theorem 5.4, we canfind an (anti-)autoequivalence Φ′ of D(X ′) which induces a stratum preserving birational mapMτ (v′) 99K Mσ′(v

′). And the composition Φ′ Φ does the work. The final part (d) ⇒ (a) istrivial.

7.4. Lagrangian Fibrations. An immediate consequence of the birationality of wall-crossingis the following result:

Theorem 7.9. Let M = Mσ(v) be a moduli space of O’Grady type, π : M →M its symplecticresolution and q(−,−) its Beauville-Bogomolov form. If there is an integral divisor class Dwith q(D) = 0 then there exists a birational moduli space M ′ = Mσ′(v

′) of O’Grady type whose

resolution M ′ admits a Lagrangian fibration.

Proof. This follows from the arguments in [BM14a, Section 11]. Indeed, we just have to replace[BM14a, Theorem 1.2] with our Theorem 7.6 in the proof of [BM14a, Theorem 1.5].

Remark 7.10. Bayer and Macrı are able to make a stronger statement in their situation if thedivisor class is also assumed to be nef; see [BM14a, Conjecture 1.4(b)]. However, for an analogousstatement to hold true for moduli spaces of O’Grady type, we would need to find a replacementfor [BM14a, Proposition 3.3] (which is a summary of Markman’s results). In particular, wewould need to know that the cone Mov(M) ∩ Pos(M) of big and movable divisors on M wasequal to the fundamental chamber of the Weyl group action on the positive cone Pos(M) of M .

Remark 7.11. There might be square-zero classes on M which are not the pullback of square-zeroclasses on M and so it is not clear whether the converse of Theorem 7.9 is true. In particular,

we see no reason why the existence of a Lagrangian fibration M → P5 should guarantee theexistence of a square-zero class on M .

Remark 7.12. A natural question at this point is whether one can combine the recent techniquesof Bayer-Hassett-Tschinkel [BHT13] and Markman [Mar11, Mar13] with similar arguments of

[BM14a, Section 12] to obtain a description of the Mori cone of the symplectic resolution M ofa moduli space M = Mσ(v) of O’Grady type. The authors plan to return to this question infuture work.

8. Examples of Movable and Nef cones

In this section, we examine examples of cones of divisors on moduli spaces of O’Grady type.

Let us suppose for simplicity that Pic(X) = Z[H] with H2 = 2d. We set v = 2vp = (2, 0,−2)and M := MH(v). Then a basis for NS(M) is given by

H = θσ(0,−H, 0) and B = θσ(−1, 0,−1)

where θσ : v⊥∼−→ NS(M) is the isometry from Corollary 2.8(3).

By Theorem 5.3, divisorial contractions are divided into two cases:



(LGU) there exists an isotropic class w with (w,v) = 2.

Just as in [BM14a, Section 13], we can solve the following system of equations

s2 = −2, (s,v) = 0 : s = (r, cH, s)

to see that the case of BN-contractions are governed by solutions to the following Pell’s equation:

x2 − dy2 = 1 via s = (x,−yH, x). (8.1)

Similarly, we can solve

w2 = 0, (w,v) = 2 : w = (r, cH, s)

to see that the case of LGU-contractions are governed by solutions to

x2 − dy2 = 1 via w =

(x+ 1

2,−yH

2,x− 1

2

)or w = (x+ 1,−yH, x) (8.2)

depending on whether y is even or odd. The two equations determine the movable cone:

Proposition 8.1. Assume Pic(X) = Z[H] with H2 = 2d. The movable cone of M = MH(2, 0,−2)has the following form:

(a) If d = k2

h2 , with k, h > 1, gcd(k, h) = 1, then

Mov(M) = 〈H, H − k

hB〉,

where q(hH − kB) = 0.

(b) If d is not a perfect square, and (8.1) has a solution, then

Mov(M) = 〈H, H − dy1

x1B〉,

where (x1, y1) is the solution to (8.1) with x1, y1 > 0, and with smallest possible x1.

(c) If d is not a perfect square, and (8.1) has no solution, then

Mov(M) = 〈H, H − dy′1

x′1B〉,

where (x′1, y′1) is the solution to (8.2) with smallest possible

y′1x′1> 0.

Proof. Setting n = 2 in the proof of [BM14a, Proposition 13.1] is sufficient here. Indeed, theirproof relies on [BM14a, Theorem 12.3] whose proof also goes through in our case when we replace[BM14a, Theorem 1.2] with Theorem 7.6.

Example 8.2. If d = 1 or 2, then we are in case (a) or (b) of Proposition 8.1 and we have

Mov(M) = 〈H, H −B〉 or Mov(M) = 〈H, H − 4

3B〉 respectively.

By Theorem 5.3, we have a flopping wall if and only if:


2 = 4.


Notice that the condition becomes 0 < (s,v) = 2(s,vp) 6 4 and so there is only enough roomfor two walls of this kind: either (s,v) = 2 or 4. Solving the following system of equations

s2 = −2, (s,v) = 2 or 4 : s = (r, cH, s)

shows that flopping walls of type (SC2) are governed by solutions of

x2 − 4dy2 = 5 and x2 − dy2 = 2 respectively. (8.3)

The associated spherical classes are s =(x+1

2 ,−yH, x−12

)and s = (x+1,−yH, x−1) respectively.

Lemma 8.3. Let M = MH(2, 0,−2). The nef cone of M has the following form:

(a) If (8.3) has no solutions, then

Nef(M) = Mov(M).

(b) Otherwise, let (x1, y1) be the positive solution of (8.3) with x1 minimal. Then

Nef(M) = 〈H, H − 2dy1

x1B〉.

where (x1, y1) is the solution to (8.1) with x1, y1 > 0, and with smallest possible x1.

Proof. We apply [BM14a, Theorem 12.1] whose proof also works in our case once we replace[BM14a, Theorems 1.2 and 5.7] with Theorems 7.6 and 5.3 respectively. The movable cone andthe nef cone agree unless there is a flopping wall, described in Theorem 5.3. By Lemma 6.13,we know that the case v = a + b with a,b positive is impossible. This leaves only the case of aspherical class s with (s,v) = 2 or 4; these exist if and only if (8.3) has a solution.

Example 8.4. If d = 1, then x2 − y2 = 2 has no solutions and the only positive solution ofx2 − 4y2 = 5 is given by (x, y) = (3, 1) which gives rise to the spherical class s = (2,−H, 1).In order to compute the generators of the nef cone of M , we need an element of v⊥ ∩ s⊥. Forexample, (2,−3H, 2) is perpendicular to both v and s and can be expressed in our basis for

NS(M) as 3H − 2B or equivalently H − 23B. In particular, we have shown that

Nef(M) = 〈H, H − 2

3B〉.

Similarly, if d = 2 then x2 − 8y2 = 5 has no solutions and the minimal solution of x2 − 2y2 = 2is given by (x, y) = (2, 1). Therefore,

Nef(M) = 〈H, H −B〉.

Remark 8.5. Observe that when (v, s) = 4, we have another spherical class t := v − s with(s, t) = 6 and so this wall is flopping the natural Lagrangian P5 ' Ext1(S, T ) inside Mσ0(v)

P5 ' PExt1(S, T ) //

Mσ0(v)

Mσ0(v − s) 'Mσ0(t) ' pt.

However, when (v, s) = 2 we have another spherical class given by t := vp−s with (s, t) = 3. Inparticular, (t,v− s) < 0 and so by Theorem 5.1, the wall is totally semistable for v− s; similararguments show that the wall is totally semistable for v − t as well. This means the floppinglocus cannot be described as a projective bundle.


Example 8.6. If X is a genus two K3 surface with Pic(X) = Z[H] and H := [C] the classof a genus two curve C, then we can twist our calculations in Example 8.4 by OX(C) to geta description for the nef cone of MH(2, 2H, 0). Moreover, since d = 1 in this case, we knowthat there is only one flopping wall corresponding to the spherical class s = (2, H, 1). That is,the movable cone has two chambers and hence Mσ(v) has two ‘birational models’ which areexchanged by the birational map Φ∗ : Mσ(2, 2H, 0) 99K MΦ(σ)(0, 2H,−2) ; E 7→ Φ(E) inducedby the spherical twist around OX .

To see that the two models really are interchanged by the spherical twist Φ, let h be the upperhalf plane, β, ω ∈ NS(X)R with ω ample and consider the open subset

h0 = h \ β + iω ∈ h : 〈exp(β + iω), s〉 = 0 when s is a (−2)-class.

If X has Picard number one then Stab†(X)/GL+2 (R) can be identified with (the universal cover

of) h0; see [BB13]. In particular, if X is a generic double cover of P2 with H2 = 2 and d = 1,then we can set β := sH and ω := tH for (s, t) ∈ R×R>0 and use the formula in [Mea12, p.35]to see that the flopping wall of Mσ(2, 2H, 0) in h0 is given by a semicircle with centre and radiusequal to (−1/2,

√5/2). Furthermore, the cohomological transform ΦH : H∗(X,Q)→ H∗(X,Q)

implies that Φ : σ0,tH 7→ σ0, 1tH and so we see that the non-Gieseker chamber with respect to

v = (2, 2H, 0) gets identified with the Gieseker chamber with respect to Φ(v) = (0, 2H,−2). If

we set H ′ := θσ(0,−H,−2) and B′ := θσ(−1,−H,−2) (to be the classes H and B twisted byOX(C)) then we can illustrate our observations as in Figure 1.

The LGU wall and hence BN wall (Lemma 6.4) can be realised geometrically by considering theobject E := Φ(Iy(−2C))[1] for some point y ∈ X with v(E) = Φ(1,−2H, 2)[1] = (2, 2H, 1) andthen splitting off another point x ∈ X. More precisely, turning the torsion sequence F → E → Oxfor F ∈MH(2, 2H, 0) yields

Ox → F [1]→ E [1].

This is Mukai’s morphism θv which contracts the projective space PExt1(E [1],Ox) ' P1. Thatis, this wall will contract a divisor of non-locally free sheaves F , and the short exact sequenceis (generically) induced by injection into a locally free sheaf E .

Remark 8.7. The only difference with [BM14a] is that we cannot say the extremal ray H−B givesrise to a Lagrangian fibration because Mσ(0, 2H,−2) is singular; we would need to precompose

with the resolution π : M →M to get this.

Remark 8.8. Recall that the six dimensional O’Grady space [O’G03] is constructed in a verysimilar way to his ten dimensional space. Indeed, he considers a moduli space of sheaves on anabelian surface, rather than a K3 surface, and takes the fibre of the Albanese map over zero.Since there are no spherical objects on an abelian surface [Bri08, Lemma 15.1], we expect thatthe analogue of our classification of walls Theorem 5.3 would become a lot simpler in this case.In particular, Yoshioka [Yos12] has already shown that all of the machinery in [BM14b, BM14a]works on abelian surfaces and so if we assume our arguments also work in this case, then wewould have to conclude that there are no flopping walls at all; c.f. [Mac12, Proposition 4.6].That is, the movable cone Mov(Mσ(v)) and the nef cone Nef(Mσ(v)) would actually coincidein this case. It seems rather surprising that the six dimensional O’Grady space Mσ(v) shouldonly have one birational model but this remark should only be regarded as speculation.

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s

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School of Mathematics, University of Edinburgh, Scotland

E-mail address: [email protected]

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom

E-mail address: [email protected]









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